Randall D.
Peters
DAMPING
PART I.
BACKGROUND AND THEORY
1. Preface
The sheer volume of published material on the subject is a testament to the difficulty of selecting topics for inclusion in a chapter on damping. Viscoelasticity alone is the basis for several voluminous engineering handbooks. The present chapter is purposely different from similarly titled chapters of other reference books. There is little repetition of well-known and proven classical methods, for which the reader is referred to excellent other sources, such as the editor's book [de Silva 2000]. They provide solution techniques for many of the routine problems of engineering. The goal of the present chapter is to provide assistance with problems that are not routine, problems that are being encountered more frequently as technology advances. It is thought that this goal is best served by revisiting fundamental issues of the physics responsible for damping.
Once a many-body system has come to steady state, its damping treatment can be far less formidable than its description during approach to steady state. When dealing with limit cycles involving aeroelasticity and joints in helicopters, nonlinearity has a profound influence in the transient behavior. Attempts to model it have been largely unsuccessful., forcing the empirical selection of elastomers to reduce the vibration. (In the old days hydraulic dampers were used.) [Hodges 2003]. At a much lower level of sophistication, our understanding of some common phenomena is quite limited, such as the negative damping character of sound generated by a violin or a clarinet. Historically, when technology 'hit the wall' because of too much theoretical 'handwaving', it was cause to examine fundamental assumptions. In physics, a complete alteration of conventional wisdom was sometimes necessary, one of the best examples being the events that gave birth to quantum mechanics. Hopefully, from the multitude of seemingly disparate (but assumed by the author as connected) observations which follow; the purpose for the architecture of this chapter can be partially realized. The enormous complexity of damping in general makes it unrealistic to hope for complete success.
Physics played a prominent role in developing
the classical foundations of damping, starting in the 19th century. Subsequently, engineers uncovered many
features of the subject that physicists never even thought about. In recent years, however, physics has been
circumstantially forced to reconsider damping fundamentals.
With the advent of
personal computing, and an increased awareness of the importance of
nonlinearity; new discoveries point to serious limitations of the classical
foundation. The field of mechanics was
severely limited until it began tackling problems of nonlinearity (not of
damping type), and became concerned with previously ignored features that can
be cause for unique system properties.
Just as these unique properties could only be solved by techniques more
sophisticated than the equations of linear type, there is mounting evidence
that nonlinear damping may be the key to understanding some bewildering
engineering cases.
It is important to try and identify the major mechanisms responsible for energy dissipation. This is easier said than done, since a host of different physics processes are usually at work. Moreover, the description of friction from first principles remains a daunting task. Thus we are forced to work with phenomenological models. There are also conflicts of nomenclature, with a given word meaning two different things from one profession to another. Thus, much of this chapter will attempt to carefully define terms while focusing on the physics, the treatment of which follows naturally along the lines of historical developments
Engineers tend to be interested in higher frequencies and higher amplitudes of vibration than are scientists. A perfect damping model would be unconcerned with such differences of application; however, such a model is far from being realized. Because small amplitude, long-period ('low and slow') oscillations provide a valuable means for studying many processes of damping in general; much of this chapter focuses in that direction.
From the multitude of choices available to writers on the subject of
damping, this author has selected a single (hopefully) unifying
theme--nonlinear damping, especially as found in low and slow
oscillations. Because it is a field
still in its infancy; many of the ideas which follow are more speculative than
one would prefer; however, they deserve discussion because of their perceived
importance. To this author's knowledge,
damping has not been previously treated in the manner of this chapter. Concerning the earliest relevant paper
[Peters, R, 2001: "Creep and ...], the following was indicated by
oft-cited Prof. A. V. Granato: "I don't know of anyone thinking about
internal friction along the lines you have mentioned."
There are two important elements to the unifying theme of nonlinear
dissipation--(i) the influence of nonlinear damping
on many-body systems in their approach to steady state, and (ii) the close
connection between damping and mechanical noise.
When vibration decay is not exponential because
of nonlinearity, there are significant ramifications, and they are only
beginning to be appreciated.
The novel features of this chapter are possible because of dramatic
improvements in both sensing and data collection/analysis in the last
decade. Demonstrating that a decay is
not purely exponential requires both (i) a good
linear sensor and (ii) the means to readily study long-time records when the
damping is small (high Q). The first
prerequisite has been met through the use of this author's patented fully
differential capacitive sensor. The
second has been realized with the availability of good, inexpensive analog to
digital converters having user friendly, yet powerful 'windows'-based
software. In addition to the 'preview'
software that comes with Dataq's A to D converter, a
proven means for identifying non-exponential decay has been the analysis of
records imported to Microsoft Excel.
Details of these novel methods will be provided in the various sections
that follow.
There are many examples in the engineering literature of nonlinear
damping; even Coulomb damping is nonlinear because the friction force involves
the algebraic sign of the velocity rather than the velocity itself, as in
linear viscous damping. What has been
evidently realized for the first time in the course of writing this chapter is
the following. As will be shown in the
material to follow, a decay process is not usually a pure exponential. Whatever the reason for a pure exponential,
whether fundamentally linear (viscous) or nonlinear (hysteretic present model),
the quality factor Q for such a pure exponential decay is constant. When there is a second mechanism such as
amplitude-dependent damping (even if it is the only mechanism), the Q now
becomes time-dependent. This is
significant to mode coupling for the following reason. When a pair of modes couple
because of elastic nonlinearity (a process that is impossible assuming linear
dynamics) the strength of the coupling is proportional to the product of the
individual amplitudes of the pair.
Consequently, variability in Q can influence the evolution to steady
state. It is a factor in determining
which modes ultimately survive and/or dominate.
Moreover, the distribution of the modes which remain depends on initial
conditions, including the intensity of excitation.
Long ago musicians learned to deal with nonlinearity, due in part to
properties of the ear that are responsible for aural harmonics. A pair of purely harmonic signals can beat in
the ear to produce a 'sound' that does not exist when sensed with a linear
detector. For example, consider a strong undistorted
500 Hz signal sounded simultaneously with a pure 1003 Hz sound. The ear will hear a 3 Hz beat due to the
superposition of the ear's aural 2nd harmonic of the first with the fundamental
of the second. But there's more to this
story. Conductors call for 'fortissimo'
and 'pianissimo' sounds, not only because of the ear's nonlinearity, but also
because of nonlinearites inherent to musical
instruments. For example, it is easy
with a good microphone and LabView to demonstrate
that the timbre of stringed instruments is intensity dependent. Not only is the mix of harmonics, as
displayed in an FFT power spectrum, different according to volume; but their
distribution also changes with time.
Noise is not typically treated in an engineering discussion of damping;
however, mechanical noise is an important part of the technical material
included in this chapter. Believing that
there is a great deal of connectivity among vibration, damping, and noise;
evidence will be provided in support of a premise -- that the most important
and least understood form of internal mechanical damping (material = hysteretic
= 'universal' ) is closely allied with the most
important and least understood form of noise (1/f = flicker = pink). If this premise is true, then the foundations
of damping physics need reconstruction on several accounts. Evidence in support of the premise will be
provided through tidbits of experimental discoveries from a host of independent
investigations. It is hoped that the
unusual and lengthy introduction which follows will be beneficial in this
regard. Historical elements serve to
synthesize the many parts and are offered without apology. Following the introduction, some practical
and novel equations of damping will eventually be provided. Even if readers find little identification
with the philosophies that birthed them, it is hoped they will at least
carefully examine the equations that are presented here in section 17 for the
first time.
2.
Introduction
2.1 General considerations of damping
The etymology of the word,
‘damping’, is difficult to determine. It is obviously allied with the word
‘damper’, commonly defined as a ‘device that decreases the amplitude of
electronic, mechanical, aerodynamic or acoustical oscillations--used for
centuries, for example, to describe the sound attenuator pedal on the
piano. Perhaps the German word, 'dampfen' (to choke), has had an influence in the evolution
of the word. One can only wonder if
water, as a moistening agent, played any role.
Certainly liquid water is important to some cases of energy dissipation
in oscillators. Moreover, friction
determined by the viscosity of a fluid (gas or liquid) is an important type of
damping. A curious piece of history, in
the celebrated work of Stokes, is why his expression, “index of friction” did
not take precedence over our modern word, ‘viscosity’. Peculiar terminology is also encountered to
describe damping, such as the engineering device known as a ‘dashpot’, which is
a mechanical damper. The vibrating part
is attached to a piston that moves in a liquid-filled chamber.
We will see that the number of adjectives used to describe various types of damping is extensive. This multiplicity of terms, to describe the loss of oscillatory energy to heat, is no doubt an indicator of the complexity of damping phenomena in general. We will attempt to (i) identify similarities and differences among various types of damping, while (ii) explaining some of the physics responsible for the characteristics observed. Conceptual ideas and techniques of both theory and experiment will be provided, targeting the lowest level of sophistication for which semi-meaningful results can be obtained. The reader should be aware that a 'grand-unified' theory of damping does not exist, nor is it likely that one will ever be created.
Damping causes
a portion of the energy of an oscillator, otherwise periodically exchanged
between potential and kinetic forms, to be irreversibly converted to heat,
sometimes by way of acoustical noise.
Whether by suitable choice of materials during design of passive
equipment, or by using feedback in active control of a sophisticated system;
control of damping is important, since mechanical vibrations can be detrimental
or even catastrophic. An oft-quoted
example of catastrophe is the Tacoma Narrows Nov. 7, 1940 . Like the vibration of a clarinet reed, this
disaster is probably best described by the term, negative damping, which can
drive parametric oscillations.
The optimal
amount of damping for a given system might fall anywhere in a wide range from
great to extremely small, depending on system needs. The engineering world frequently wants
oscillations to be as close to critically damped as possible. Physics experiments such as those searching
for the elusive gravitational wave [centered at the Laser Interferometer
Gravitational Observatories, or LIGO, in the United States; GEO600 in Germany Italy Japan
For the
specific components of a system, a successful design frequently requires
identification of the specific mechanisms primarily responsible for the
dissipation of energy. Even after
identifying the dominant sources, the theoretical difficulty of their treatment
can also range from great to small, depending on the type of damping. For dashpot fluid damping, adequate models
have existed for decades. For material
damping, on the other hand, theories of internal friction are numerous and
largely lacking in self consistency.
The fundamental
mechanisms responsible for damping are in most cases nonlinear; however, the oscillator's
motion can itself be approximated in many cases by a linear 2nd order
differential equation. If the potential
energy is quadratic in the displacement, then the undamped
linear equation of motion is that of the simple harmonic oscillator; because
its solution is a combination of the sine and cosine (harmonic) functions. This undamped
equation comprises the sum of two terms, one being a displacement, and the
other term an acceleration. The constant parameter multiplying each term
of the pair depends on the nature of the system. For example, in the case of a mass-spring
oscillator, the acceleration is multiplied by the mass, and the displacement by
the spring constant. Thus the equation
corresponds to Newton
The usual means
to describe damping, which is always present with oscillation, is to add a
velocity term to the aforementioned displacement and acceleration. Although the damping could derive from several
causes, there
is usually a single dominant process.
For example, the damping of current in a series-connected RLC circuit
may depend mostly on Joule heating in the resistor R; in spite of the fact that
there must also be energy loss in the form of radiation. Thus the equation of motion includes a first
time derivative of the capacitor's charge (current) multiplied by R, in accord
with Ohm's law.
Whether
radiation is important for damping of the RLC circuit depends on the amount of
coupling to the environment. If the
circuit communicates with a final amplifier connected to an antenna; then
radiation may become more important than Joule losses. The frequency of oscillation is a key
parameter in this case, and also for damping problems in general. Unfortunately for some common systems,
theoretical efforts to properly account for the effects of frequency have
proven largely unsuccessful--except for models of phenomenological type,
developed by empiricism.
2.2 Specific considerations
The mass-spring oscillator is the textbook example of harmonic motion, for which one of the most sophisticated mechanical oscillators ever built is the LaCoste version of vertical seismometer. Significant portions of the experimental data presented in this chapter were generated with an instrument designed around the LaCoste zero-length spring [LaCoste 1934]. The instrument used for this data collection was part of the World Wide Standardized Seismograph Network (WWSSN) during the 1960's. The spring of this seismometer is responsible for hysteretic damping of the instrument, rather than viscous damping as commonly assumed. Contrary to popular belief, air damping is not important for this seismograph at its nominal operating period, which is typically greater than 15 s. Since every long-period pendulum apparently exhibits similar behavior, we thus find strong synergetic evidence in support of an old (mostly unheeded) claim that hysteretic damping (friction force independent of frequency) is universal [Kimball 1927]. Their claim in 1927 to have discovered a universal form of internal friction (damping) is strengthened -- since the same behavior is seen in three distinctly different systems: (i) a mass-spring oscillator (as demonstrated by Gunar Streckeisen, details given later), (ii) a pendulum whose restoration depends on the Earth's gravitational field (demonstrated by several independent groups), and (iii) a rotating rod strained by a transverse deflection (1927 experiments of Kimball and Lovell).
The assumption of universality for hysteretic damping is a key point of this chapter. It will be shown that the damping of even a vibrating gas column (Ruchhardt's experiment to measure the ratio of heat capacities) is likely also hysteretic. The models that are described represent a departure from common theories of damping. Interestingly, the author's model has similarities to ordinary sliding friction, as given to us by Charles Augustin Coulomb. It effectively modifies the Coulomb coefficient of kinetic friction to yield an effective energy -dependent internal friction coefficient. The energy dependence is necessary to obtain exponential decay, as opposed to the linear decay of Coulomb damping. Just as with conventional Coulomb damping, its form is nonlinear, involving the algebraic sign of the velocity. We will see that the damping capacity predicted by the model permits an equivalent viscous form. Yet the underlying physics is related to creep of secondary type as opposed to the primary creep of viscoelasticity.
It is this author's opinion that much of the existing theory of damping is not the best means for modelling dissipation. The difficulties arise from approximating oscillator decay with linear mathematics. Although most individuals recognize the oft-stated caveat that viscous damping is an approximation to the actual physics of dissipation; they do not recognize some of the many serious limitations of the approximation. The situation is similar to the place in which we found ourselves at the beginning of the era labeled 'deterministic chaos'. The 'butterfly effect' [Lorenz 1972] has radically altered the thinking of many, but only in relationship to large amplitude motions of a pendulum, where the instrument is no longer iscochronous because of nonlinearity. As an archetype of chaos, the pendulum must be rigid and capable of 'winding' (displacement greater than p) before chaos is possible. Nonlinearity is a prerequisite for the chaos, but it is not sufficient; since there are many examples of highly nonlinear but non-chaotic motions. For example, amplitude jumps of nonlinear oscillators, during frequency sweep of an external drive, have been known for many years. They were observed before chaos was recognized, in systems like the Duffing and Van der Pol oscillators. Yet chaos, with its sensitive dependence on initial conditions (responsible for the butterfly effect), was not contemplated at the time. As with most significant advances, Lorenz's discovery was by accident, as he modelled convection in the atmosphere. The author's confrontation with complexity that derives from mesoscale structures in metals was likewise unexpected. "Strange phenomena" (as Richard Feynman would probably have labeled them) were encountered while using his patented fully differential capacitive sensor to study various mechanical systems, mainly oscillators.
As with chaos, the pendulum may ultimately serve as an archetype of complexity. When operated at low energy, especially through a combination of long period and small amplitude; the free-decay of the physical pendulum departs radically from the predictions based on linear equations of motion. Such complexity can be easily demonstrated when the pendulum is fabricated from soft alloy metals. For example, Fig. 1 illustrates the decay of a 'rod pendulum' constructed with ordinary (heavy gauge lead-tin) solder of the type used for joining electrical conductors [Peters 2002: ".. the 21st century]
Figure
1.
Free-decay of a rod pendulum fabricated from solder
The 'jerkiness' (discontinuities) in the record of Fig. 1 is in no way
related to amplitude jumps of the type previously mentioned; rather these are
jumps of the Portevin LeChatelier (PLC) type [Portevin 1923]. They
are a fundamental, yet 'dirty' phenomenon that physics has chosen for
decades to try and ignore (even though materials science and engineering took
early note of the PLC effect ). The most
obvious and profound thing that can be said about Fig. 1 is the following: the presence of PLC jerkiness means that the
concept of a potential energy function is not really valid, since the
requirement for its definition is that a closed integral of the force with
respect to displacement must vanish.
No matter the form of hysteresis, which is the
cause for damping, it disallows the curl of the force to be zero; so that
potential energy is never formally meaningful for a macroscopic oscillator
(since there is always damping). In
those cases where the damping is essentially continuous (not true for the
example of Fig. 1), the assumption of a potential energy function retains some
computational meaning. For oscillators
influenced by the PLC effect, this is no longer true. The resulting properties
are important to a variety of technology issues, such as sensor performance;
since noise is no longer the simple thermal form predicted by the
fluctuation-dissipation theorem (used to characterize white; i.e, Johnson noise).
Practical means for dealing with
systems influenced by 'stiction' have been known by
engineers for decades. Because of metastabilities
in the assumed potential function, the system is prone to 'latching' (stuck in
a localized potential well). One means
for mastering the metastabilities (unstick the part designed to move) is to 'dither' the
system. The process has become more
sophisticated in the last decade, which saw a major growth of interest in
'stochastic resonance'. In the definition by Bulsara and Gammaitoni [Bulsara 1996]. "A stochastic resonance is a phenomenon
in which a nonlinear system is subjected to a periodic modulated signal so weak
as to be normally undetectable, but it becomes detectable due to resonance
between the weak deterministic signal and stochastic noise." The phenomenon is related to dithering [Gammaitoni 1995].
It is a case where SNR can be increased by the counterintuitive act of
raising the level of
noise. Such a gain in SNR
is not possible with a harmonic potential.
In recent studies of granular materials, 'tapping' has become a popular
means to study behavior that violates the fundamental theorem of calculus. Years ago this author used tapping as a
means to accelerate creep in wires under tension. Evidently, hammering the table on which the
extensometer rested, caused vibrational
excitations of the wire that stimulated length changes of discontinuous PLC
type. Because of the broad spectral
character of an impulse, various eigenmodes of the
wire could be thus readily excited.
After 'hammering-down' under load, a silver wire could by this same
means be stimulated, after partial load removal, to exhibit length
contractions. Since the total number of
atoms is fixed, the process must involve exchange of atoms between the surface
of the sample and its volume.
Extensometer studies of wires at
elevated temperature have also displayed strange behavior. A polycrystalline silver wire of diameter 0.1
mm and approximate length 30 cm was found to exhibit large fluctuations when
heated in air to within 100K of its melting point, using a vertical furnace
[Peters 1993, Fluctuations ..]. The large fluctuation in length at these
temperatures (reminiscent of critical phenomena and visible to the naked eye)
may be associated with oxide states of the metal, since the experiments were
not performed in vacuum. When cycled in
temperature, fluctuations in the length of a gold wire were found to exhibit
dramatic hysteresis.
With influence from Prof. Tom Erber of
Mechanical hysteresis resulting from mesoanelastic defect structures is evidently ubiquitous. Piezotranslators, which are used as actuators in atomic force microscopes and other nanotechnology applications, are afflicted with high levels of hysteresis when operating open loop. This behavior is consistent with the anomalously large damping that was observed with a pendulum (reported elsewhere in this document) in which there was a steel/PZT interface for the knife-edge.
Even the common strain gauge exhibits complex hysteresis behavior. The normally large hysteresis that is observed in preliminary cycling of a gauge is typically reduced by significant amounts after repeated cycling, a type of work hardening (if strained well below failure limits). It should be noted that the hysteresis of all the discussions in this chapter is not to be confused with backlash (as in a gear train).
All of these experiments are in keeping with the premise that mesoanelastic complexity determines the nature of hysteretic damping. It is seen that there are a plethora of examples where strain (and thus damping) of a sample is not simple, is not smooth -- but more like the complex behavior of granular materials. In the case of polycrystalline metals, the same grains that are made visible by methods of acid etching decoration--are evidently responsible for mesoscale (non-smooth) internal friction damping. To assume that damping is quantal at the atomic, rather than the mesoscale, is without experimental justification. Nevertheless, this is a popular assumption with which estimates of the noise floor of an instrument such as a seismometer is estimated, to calculate signal to noise ratios (SNR).
2.3 The pendulum as an instrument for the study of material damping
Because of its early contributions to physics,
which in those days was called natural philosophy; one might be tempted to
believe that the pendulum is only important to (i)
the history of science, or (ii) teaching of fundamental principles. A single
observation should be sufficient to resist this temptation-- (as already noted)
the pendulum has in the last thirty years become the primary archetype for the
new science of chaos. Additionally, many of the data sets of this document, which
show significant and previously unpublished results--
were generated with a pendulum.
To a student of elementary physics, the choice of a pendulum may seem
unsophisticated. Yet to the author, who
has spent 15 intense years trying to understand harmonic oscillators, the
pendulum is the most versatile instrument with which to understand
damping. It has been central to the
development of science in general. It
was studied by Galileo, Huygens,
In 1850 Sir George Gabriel Stokes published a foundational paper [Stokes 1850]. His treatment of pendulum damping permitted the understanding, decades later, of a number of important phenomena in physics and engineering. For example, his studies were foundational to the Navier-Stokes equations of fluid mechanics. Moreover, viscous flow known as Stokes' law was the basis for Millikan's famous oil drop experiment that determined the charge of the electron.
Stokes noted in his paper that, "...pendulum observations may justly be ranked among those most distinguished by modern exactness". He also noted "The present paper contains one or two applicatons of the theory of internal friction to problems which are of some interest, but which do not relate to pendulums. ....... the resistance thus determined proves to be proportional, for a given fluid and a given velocity, not to the surface, but to the radius of the sphere. ... Since the index of friction of air is known from pendulum experiments, we may easily calculate the terminal velocity of a globule [water] of given size. ....The pendulum thus, in addition to its other uses, affords us some interesting information relating to the department of meteorology. " The last statement of this quote speaks to some of the errors in the 'common theory' of his day. In similar manner, some of the common-to-physics damping models of today are erroneously applied. Those who hold the viscous damping linear model in unwarranted regard, fail to recognize the limitations under which it is valid. There are frequent misapplications for reason of experimental deficiencies. We can all profit by taking seriously the following well known words of Kelvin:
"When you can measure what you are
speaking about, and express it in numbers, you know something about it. But when you cannot measure it, when you
cannot express it in numbers, your knowledge is of a meagre
and unsatisfactory kind. It may be the
beginning of knowledge but you have scarcely in your Thoughts advanced to the
state of science." - William
Thomson, Lord Kelvin, 1824-1907.
Simple (viscous) flow of the Stokes law type is possible only according to the restrictive conditions that Stokes spelled out in his paper. We now specify those conditions for viscous flow according to the nondimensional parameter given us late in the 19th century by Osborne Reynolds. Specifically, Stokes law is valid only for Re = rvL/h < 60 (approximately, for spheres); where r is the density of the retarding fluid, v is the speed of the object relative to the fluid, L is a characteristic dimension of the object, and h is the viscosity of the fluid. The requirement is not generally met for oscillators, and recent experiments have shown that contributions to the damping from air drag proportional to the square of the velocity cannot generally be ignored [Nelson 1986]. This is just one example of how two or more damping types must sometimes be folded into an adequate model of dissipation. A novel method for combining all the common forms of damping in one mathematical expression is provided in this document. Additionally, it is shown how to analytically calculate the history of the amplitude of free-decay for such cases.
Considering the importance of Stokes' work, it is surprising that some of his requests for further experiments were apparently never seriously considered. On page 75 of his paper one reads the following: "Moreover, experiments on the decrement of the arc of vibration are almost wholly wanting." Having noted this, Stokes appealed to experimentalists to generate such data. In the 19th century, collecting the data he requested would have been labor intensive and therefore the experiments were probably never attempted. Sensors and data processing of the modern age now make them straightforward, but the pendulum has by now been viewed by too many as a relic; rather than the important instrument described by Stokes. Much of the author's efforts have been directed at showing that the pendulum is still an important research instrument. For example, one physical pendulum of simple design was the basis for the generalized model of damping (modified Coulomb) that is here presented. Another has been used to illustrate surprisingly rich complexities of the motion that result from the ubiquitous defects of its structure[Peters 2002 ...21st ..]. Thus studying the complex motions of 'low and slow' physical pendula could yield significant new insight into the defect properties of materials--a field where relatively little first-principles progress has been made.
2.4 "Plenty of Room at the Bottom"
Richard Feynman gave a now-famous talk in 1959 titled, "There's plenty of room at the bottom" [presented at the American Physical Society's annual meeting at CalTech]. Drawing on observations from biology, he spoke of a solid state physics world involving "...strange phenomena that occur in complex situations." In the 44 years since Feynman's prophetic comments, there have been spectacular achievements in VLSI electronics, microelectromechanical systems (MEMS), and even nanotechnology. Progress in the mechanical (including sensor) realm has been much slower than in electronics; consequently, our present processing power far exceeds our acquisition (and actuator) capabilities.
One of the major obstacles to miniaturization involves dramatic change to physical properties that can occur as the size of a system shrinks below the mesoscale toward the atomic. For example, VLSI electronics is already beginning to be impacted by quantum properties of the atom, as component size continues to decrease in accord with Moore's law [Gordon Moore, 1965] Among other things, Feynman predicted that lubrication would no longer be 'classical' at such a scale. On a related note, a paper by Nobel Laureate Edward M. Purcell [Purcell 1977] draws a striking contrast between our macroscopic world and that of micro-organisms. At low Reynolds number inertia becomes unimportant, and mechanics is dominated by viscous effects. The adoption of a new paradigm will be necessary for engineers to deal with these differences.
In the article, "Plenty of room indeed" [Roukes 2001] it is noted that there is an anticipated 'dark side' of efforts to build truly useful micro- and nano-sized devices. Gaseous atoms and molecules constantly adsorb and desorb from device surfaces. This process is known to exchange momentum with the surface, even permitting scientific study of the gas-solid interface [Peters 1990, Desorption...]. The smaller the device, the less stable it will be because of adsorption/desorption. As Roukes has noted, this instability may pose a real disadvantage of various futuristic electromechanical signal processing applications [Cleland 2002].
There is direct evidence provided in the present chapter that we need to be more concerned with noise: (i) the evacuated pendulum where it is speculated that outgassing influenced its free-decay, and (ii) the seismometer free decay that showed both amplitude & phase noise and evidence for nonlinear damping. Concerning (i), when the vacuum chamber pressure is reduced, the pre-existing steady state (normal rate balance between adsorption and desorption) becomes disturbed; so that there is a complex emission of gases from the surface of the pendulum. The emission is not likely to be spatially uniform, but more like the jets seen on Halley's comet when photographed by the Giotto spacecraft in 1986. In case (ii), the noise is seeen to derive from mesoanelastic complexity of the structure of the pendulum itself rather than involving gases.
Miniaturized devices have the
potential to serve as on-chip clocks; and the importance of phase noise to
clocks is well documented. There is
another, more subtle issue that points out the importance of phase noise. One of the best means for improving SNR is
the technique of phase sensitive detection, first employed by Robert Dicke at
Phase noise of miniaturized devices is still mostly speculative. In addition to the mechanism just mentioned (adsorption/desorption) there is the matter which constitutes the theme of this chapter, defect organization. It is not possible to grow materials without dislocations and/or other disturbances to crystalline order, such as vacancies, interstitials, or substitutional impurities. Thus, "when mother nature fills the vacuum she abhors, she rarely does so with perfection".
Long before defects organize to the point of incipient failure (at much larger strains), they still influence vibration. They may even be a primary source of 1/f noise. Electronic noise of 1/f type is known to involve defects by means of trapping states, and these states derive from crystalline defects sometimes involving the surface. The interaction of the surface and the volume of a solid is important. For example, consider pure copper single crystals of the type used by the author in his doctoral work. A practical joke suggested by Vic Pare' (that we never conducted because of the cost of these samples) would be to have a '98 lb weakling' bend one by hand; then ask an NFL linebacker to straighten it back out! The striking irreversibility is the result of work hardening--as dislocations develop at the surface and propagate into the bulk where they entangle.
In the case of polycrystalline materials, the memory features of hysteresis may be important according to the method of their fabrication. Wires are typically produced by pulling through successively smaller dies. This 'swaging' may be conducive to the exchange of monolayer groups of atoms between the volume and the surface during fluctuation length changes. The fluctuation-dissipation theorem does not hold; or if it does, only in terms of larger entities than the atom. Thus there are many 'yet to be quantified' elements of noise in the vibration of miniaturized devices. Feynman was right when he spoke of strange phenomena of the solid state.
Technology of the future is expected to be confronted increasingly with damping problems that must address issues of scaling--to deal with some factors discussed in this chapter; which to the author's knowledge, have not been previously published. Until small (MEMS) oscillators become more common to the engineering world, we must study the mechanisms responsible for their damping by other than traditional means. One approach is similar to experimental techniques for the verification of the kinetic theory of gases. As noted by Present in his textbook [Present 1958], there are two ways that Brownian motion can be studied: either (i) with small objects and an unsophisticated detector, or (ii) with larger objects and a very sensitive detector. It is the latter that provided some of our present knowledge of damping at the mesoscale. The fully differential capacitive transducer whose patent label is "symmetric differential capacitive" (SDC), is a robust new technology that is sensitive, linear, and user friendly [Peters 1993, ``Full-bridge..]. As with other sensitive detectors that have been used to predict the properties of small objects by studying larger ones, small-energy studies of various macroscopic pendula are demonstrating some of the 'strange phenomena' of complex type predicted by Feynman.
From the author's perspective, we of the physics community have been guilty of two significant errors: (i) oversimplification of many problems by assuming a linear equation of motion based on viscous damping, or (ii) losing sight of fundamental issues by working with inappropriate, overly complicated damping models. The goal of this chapter is to assist progress toward a healthier balance between these extremes. It is hoped that readers will be thus better equipped to identify, and then dismantle some of the impediments to the development of future technologies.
3. Background--
3.1 Terminology
The large number of mechanisms capable of energy dissipation has resulted in a host of adjectives to describe damping phenomena in mechanical systems. They include (non-exhaustive list) viscous, eddy current, Coulomb, sliding, friction, structural, fluid, thermoelastic, internal friction, visoelastic, material, solid, phonon-phonon, phonon-electron, and hysteretic. For present purposes, damping types will be grouped according to one of the following three categories: (i) fluid (including viscous), (i) Coulomb, and (iii) hysteretic. Although hysteretic damping has come in engineering circles to be associated with a particular form of material damping in solids, it should be noted that all forms of damping involve hysteresis (for which the Greek meaning of the word is "to come late"). In a plot of periodic stress versus strain, which is a straight line for displacements of a non-dissipative, idealized substance; hysteresis causes the line to open into a loop. The size of the loop, more specifically the area inside this hysteresis loop, is a measure of the amount of non-recoverable work done per cycle because of the damping. An actual force of friction is not readily recognizable in those cases that are labeled 'internal friction'. The word 'friction' is used in a generic sense, meaning any process responsible for conversion of the oscillator's coherent motion into incoherent thermal activity.
With each of viscous, eddy current, and Coulomb
damping, a force external to the oscillator is responsible for the dissipation
of energy. The external force is
associated respectively with (i) laminar fluid flow,
(ii) induced currents, and (iii) surface friction. The surface friction case is not necessarily
the trivial textbook presentation involving a coefficient of kinetic friction
and a normal force. The cases just
mentioned, along with thermoelastic damping; which is
of internal rather than external origin, are much easier to treat theoretically
than other cases. Viscous damping and
eddy current damping (over the full range of the motion) are adequately
described by a velocity term, which yields a linear equation of motion. Coulomb damping, however, is not proportional
to velocity, but rather depends only on the algebraic sign of the
velocity. The equation of motion is
consequently nonlinear. Additionally,
and unlike most other forms, Coulomb damping is not exponential. The turning points lie along a straight line,
when the motion is plotted versus time.
Similarly, if eddy current damping exists only over a small part of the
motion, the decay is linear rather than exponential [Singh 2002].
3.2 General technical features
Historically, viscous damping has been the model of choice, because the resulting equation of motion is mathematically attractive; and for the RLC circuit, the form is appropriate. For mechanical oscillators, it is not generally appropriate, since viscous damping amounts to some part of the system moving in an external Newtonian fluid that removes energy because of a friction force that is proportional to velocity.
The defects responsible for material damping, such as dislocations, are also responsible for creep; so that high strength and high damping tend to be incompatible attributes. Magnesium alloys tend to be better than many other metals in this regard. Hardness of a material is neither a prerequisite for toughness nor for small damping, as recognized by those familiar with the mechanical properties of cast iron.
On a different
scale, defects determine "how things break"; concerning which Marder
and Fineberg have stated the following: "the
strength of solids calculated from an excessively idealized starting point
comes out completely wrong; it is not determined by performance under ideal
conditions, but instead by the survival of the most vulnerable spot under the
most adverse of conditions." [Marder 1996]
Three famous
scientists are primarily responsible for the highly popular viscous damping
model of the simple harmonic oscillator; they are Lord Kelvin [Thomson 1873],
G. G. Stokes and H. A. Lorentz. Stokes is best known for his equations of
fluid dynamics that also include the name Navier. Stokes’ law, which describes the terminal velocity
of a raindrop, was developed through his treatment of the damping of a
pendulum. Not only does his law provide
a basis for the simplest approximation for damping of a macroscopic oscillator , it was used by Robert Millikan
to determine the charge of the electron.
It should be noted that harmonic oscillation in a fluid (even at low
Reynolds number) is much more complicated than steady flow visous
friction. This topic is treated in PART
II, Section 9.
The first individual to use the term, "simple harmonic oscillator" was probably Lord Kelvin. Such an oscillator is a key tool of experimental physics and also the foundation for much of theoretical physics. It is the basis for communication via electromagnetic waves and even esoteric theories of superfluids and superconductors.
Much of the underpinnnings of theory involving harmonic oscillation derive from the work of Hendrik Anton Lorentz (1853-1928). Lorentz is well known for a variety of classical physics contributions, such as (i) the transformation of special relativity associated with Einstein , and (ii) the force law for the acceleration of charged particles—both of which bear his name. Before the existence of electrons was proved, Lorentz proposed that light waves were due to oscillations of an electric charge in the atom. For his development of a mathematical theory of the electron, he received the Nobel Prize in 1902. The importance of his contributions is further realized by noting that it is common practice to describe the lineshape of atomic spectra by the term, ‘Lorentzian’. The Lorentzian is equivalent to the resonance response of the driven viscous-damped simple harmonic oscillator.
It is easy to
show how resistance in an electric network is responsible for damping; however,
it is a challenge to understand anelastic processes
of mechanical damping in terms of viscosity.
From comments of his PhD dissertation, it has been said that even Lorentz was never apparently satisfied with the velocity
damping term in his equation—not knowing just how to relate it to the
underlying physics. It is also clear
from Stokes’ paper that he recognized the need for caution in the use of his
law of viscous friction. It appears that
both Lorentz and Stokes were very careful, compared
to the carelessness with which the viscous model has been employed by many
individuals in recent years.
The failure of
solids influenced by ‘hysteretic’ damping to be adequately described by the
methods of viscoelasticity is not widely
appreciated. It is unfortunate that too
few people have expanded their view of damping to include other important
types, such as derive from the anelasticity of
solids. It is important in this work to
recognize some subtle differences; for example inelastic (not elastic) is not
to be equated with anelastic (other than elastic).
3.3 Active versus Passive Damping
With improvements in cost/performance of electronics, active damping is increasingly popular. Using force-feedback with integration/differentiation circuitry (opamps), a mechanical oscillator can sometimes be tailored for a specific purpose. A sophisticated example of this technology are the broadband seismometers that began to replace earlier version (passive) instruments, roughly thirty five years ago. The Sprengnether LaCoste-spring instrument that was used for some of the experiments reported in this document have been superceded by force-feedback units such as the Streckeisen STS-1 and STS-2.
In lieu of feedback, another way to
actively influence the damping of a mechanical oscillator is to connect the
sensor to an amplifier having a negative input resistance. The seismometers marketed by Lennartz Electronics in
Active damping depends on the nature of the transfer function of the composite system (electronics plus mechanical). The characteristics of the transfer function are determined by the location of its poles and zeros in the complex plane. Seismometers operate nominally near 0.707 of critical damping. This is done for two reasons: (i) the instrument is easier to adjust, and (ii) the interpretation of earthquake records is simpler. Of course, to increase damping is to decrease sensitivity because of the fluctuation-dissipation theorem.
The force-feedback technique is not practical for some situations, regardless of cost. Additionally, it must be recognized that the method is not the answer to all problems, since electronics cannot compensate for a poor mechanical design. The description of commercial products is in some cases highly exaggerated, giving the impression that most any sensor can perform flawlessly in this manner. Some accelerometers have employed dithering to offset the effects of 'stiction' in bearings. The dithering was necessary because the potential energy function is not truly harmonic, being afflicted with the consequences of nonlinear damping. Even with sensing schemes that do not use a bearing, the effects of nonlinearity persist. In "Seismic Sensors and their Calibration" [ed. Bormann 2002] Erhard Wielandt, in talking about transient disturbances in the spring of a seismometer, says the following: "Most new seismometers produce spontaneous transient disturbances, quasi-miniature earthquakes caused by stress in the mechanical components." In other words, internal friction from defects at the mesoscale cause behavior that is in some ways similar to ordinary sliding friction, where the static coefficient is greater than the kinetic coefficient. The postulate of Bantel & Newman is consistent with this idea [Bantel 2000] when they refer to their observations as being consistent with a 'stick-slip' model of internal friction.
It is seen then that one must use a detector that responds faithfully to the signal around which the servo-network functions. The linearity and sensitivity of that sensor are of paramount importance, since the basis for force-feedback design is linear system theory. For some less challenging cases the design approach is straightforward, since software packages like MatLab have built in functions to describe behavior.
3.4 Magnetorheological (MR) Damping
A recent approach to damping control, that is quite different from the servo-networks mentioned above, is one which uses an MR fluid. It takes advantage of the large variation in viscosity of certain compound fluids according to the size of an applied magnetic field. J. David Carlson [Carlson 2002] describes how an MR sponge damper is activated during the spin cycle of a washing machine to keep it from 'walking out of the room'. The peak in the Lorentzian (resonance response) of the machine is shown in his article to be substantially lowered by supplying current to the electromagnet of the damper.
3.5 Portevin LeChatelier (PLC) Effect
Physics, engineering, geoscience, and mathematics have all contributed greatly to a better understanding of damping phenomena; however, there has been little cross-discipline exchange of ideas and lessons learned. Some of the impediments to strong interdisciplinary programs derive from (i) the complexity of damping problems in general, and (ii) the tendency for physics and mathematics research to be on the one hand less pragmatic, and on the other hand highly specialized--focused on specific energy dissipation mechanisms. A good example of (i) involves the PLC effect, discovered in 1923. Why physics mostly ignored this early example of 'dirty science' by two of their own number is not easily understood, although the birthing of quantum mechanics around this time may have been a factor. Had history turned a different direction, perhaps we would already be able to explain from first principles the most important, but still barely-understood form of noise, known as 1/f , or flicker, or pink noise. Even though R. B. Johnson (well known for his discovery of white electronics noise in a resistor) was one of the first to see this form of noise, it still is not explained from first principles--although recent discoveries suggest an intimate connection with fractal geometry involving self-similarity. Such geometry is associated with the mesoscale of materials, where the grain, rather than the atom, is the basic element of statistical mechanics, rather than the atom.
For alloys, the PLC effect appears to be in some ways what the Barkhausen effect is to magnetic systems. In the case of ferrous materials, the noise which derives from the mesoscale has been long recognized; however, similar noise of mechanical type has not been seriously studied. This oversight is even more puzzling when one considers the admonition by G. Venkataraman, as recorded in the proceedings of a Fermi conference, for scientists to get involved in what he felt should become an important new field [Venkataraman 1982].
3.6
Noise
Noise is purposely discussed in this chapter , because it has been a largely-missing component of efforts to understand the physics of damping. A feel for the importance of noise to damping research is to be gleaned from a comment by Kip Thorne in his foreward to the English translation of a book by V. B. Braginsky et al [Braginsky 1985]. Mainly because of instrumental needs of the Laser Interferometer Gravitational Wave Observatories (LIGO), Thorne writes, "The central problem of such experiments is to construct an oscillator that is as perfectly simple harmonic as possible, and the largest obstacle to such construction is the oscillator's dissipation. If dissipation were perfectly smooth, it would not be much of an obstacle, but the fluctuation-dissipation theorem of statistical mechanics guarantees that any dissipation is accompanied by fluctuating forces. The stronger the dissipation, the larger the fluctuating forces, and the more seriously they mask the signals that the experimenter seeks to detect."
This comment by Thorne suggests a frequently important impediment to dialogue between engineering and physics--concern for different issues. LIGO is trying to minimize damping, whereas many engineering problems are concerned with just the opposite--making the damping as large as possible without compromising strength. More detailed discussions of noise are provided later.
3.7 Viscoelasticity
Within the world of polymers,
damping is frequently described by the expression 'viscoelasticity'. This word, around which handbooks have been
written [
It should be noted that the springs and dashpots used in models of viscoelasticity do not actually exist. They serve as a phenomenological means for (hopefully) understanding the elementary processes which their arrangement is designed to mimic. Consider, for example, high polymers, in which the interwoven structure of the long-chain molecules is one of extensive mechanical interference. (One popular visualization is that of an entanglement of a huge number of long, writhing snakes.) An increase of temperature is met with overall length reduction (negative temperature coefficient of expansion for the so-called entropy spring), which stands in stark contrast with metals. Such behavior is clearly important to damping, since as noted by Gross years ago ".... thermal movement interferes with the orientation and disorientation of the molecules and ultimately causes delay in the expansion and contraction of the specimen" [Gross 1952].
3.8 Memory Effects
In this same article, Gross is one of the first to mention 'memory' properties of creep. He describes a by-then old demonstration in which a "firmly suspended metal or plastic wire is twisted first in one direction for a long time and then in the other direction for a short time. Immediately after release, the deflection will be in the direction of the last twisting, but it decreases rapidly. Presently, a reversal occurs, and the wire begins to turn in the other direction, corresponding to the first twisting--the memory of the recent short-term handling has been obliterated by that of the more remote but longer lasting and therefore more impressive one!" Perhaps this old demonstration (sometimes today called the anelastic after-effect) is not so startling to those familiar with more modern shape-memory-alloys, which are expected by many to play increasingly important roles in the applied science of damping.
3.9 Early history of viscoelasticity
Those who
provided seminal influence in the development of the theories of viscoelasticity during the 19th century were some of the
most famous names in physics, like Maxwell and Kelvin. Maxwell is best known
for the electromagnetic equations associated with his name. He is far less known for two other
significant contributions: (i) kinetic theory of gases and (ii) viscoelasticity—both
of which are important to theories of oscillator damping. Maxwell’s interest in the problem of viscoelasticity is first documented in a paper during his
teen years, titled “The Equilibrium of Elastic Solids”. Through his development
with Boltzmann of the kinetic theory of gases, Maxwell showed a
counterintuitive property of the viscosity of a gas. The viscosity does not decrease significantly
as the pressure is reduced, until the mean free path between collisions of the
molecules begins to approach dimensions of the chamber holding the gas. Important even to modern innovations such as
MEMS oscillators, his surprising prediction was quickly verified by
experiment. Maxwell's model of viscoelasticity combines a purely elastic spring with a
purely viscous dashpot (fluid damper in which the friction force is
proportional to the velocity).
Kelvin,
probably the first to include a viscous damping term in the equation of motion
of the simple harmonic oscillator, developed a similar model of viscoelasticity.
Each of the two models is usually represented in literature (without
original references) as containing a single spring and a single dashpot. They differ in that one connects the pair in
series (Maxwell), while the other connects them in parallel
(Kelvin-Voigt).
Both the Maxwell model and the Kelvin-Voigt model have been found by engineers to be less useful than the standard linear model (SLM) of anelasticity, largely advanced in the 20th century by Clarence Zener [Zener 1948]. In the three component Zener model, a spring is connected in series with a parallel combination of spring and dashpot. Curiously, Zener is widely associated with electronics because of the common diode named after him, but fewer people know of his work in anelasticity. No doubt, his understanding of anelasticity helped him to better understand the complex processes at work in his diode.
3.10
Creep
The prevailing models of anelasticity appear frequently in the literature, but mostly in relationship to primary creep. Some of the papers exceptional to this rule are those by Berdichevsky [Berdichevsky 1997]. Recent work of a more heuristic type has shown that the equations of viscoelasticity are also able to accommodate secondary creep, in which the decay of strain rate with time has disappeared [Peters 2001, "Creep ....] .
The importance of creep (and relaxation) physics to damping warrants some discussion. When a sample is subjected to a constant stress, the strain evolves through three phases of creep: (i) primary, (ii) secondary, and (iii) tertiary. An example of the first two of these phases is shown in Fig. 2.
Figure 2. Example of creep in the spring
of a vertical seismometer
In the primary stage, the sample is deformed by anelastic processes involving defects of the crystalline structure. Influence of the disordering mechanisms is progressively reduced as the sample undergoes work hardening (such as pinning of dislocations). Work hardening would result in a purely exponential creep, in the absence of thermal effects, which strive to undo the hardening (via diffusion processes). (At zero Kelvin, the creep would eventually cease, if described by a single time constant.) In the secondary stage, a balance between work hardening and thermal softening is attained, in which the strain versus time has converted from exponential to linear. This balance cannot continue forever, if the stress is larger than a threshold associated with failure (the elastic limit); and thus a final complex fracture of the sample finally occurs as the sample passes through the tertiary stage. Although one might want to divorce the issues of tertiary creep from considerations of damping, there is clearly a link between damping and failure, involving defects. We will return to this point later.
In Fig. 2 creep resulted from the instrument having been severely disturbed (relocated accidentally by plumbers working in the building). As with any long period mechanical oscillator, it is necessary for this instrument to stabilize after a major rebalancing. Primary creep is seen to have endured for about 5 hours and what is labeled secondary creep in the figure doesn't continue indefinitely with the implied constant rate. Thus the instrument will typically stabilize after one or several days, when the period of oscillation is of the order of 20 s.
The total amount of creep in Fig. 2 deserves mention. In the indicated 14.2 hours, the mass of the seismometer moved a vertical distance of only 1/4 mm, which can be ascertained from the ordinate axis, using the sensor calibration constant of 2000 V/m .
3.11 Stretched exponentials
Systems typically demonstrate a more complex behavior than can be simply described by the standard linear model of viscoelasticity. In 1847, Kohlrausch [Kohlrausch 1847] discovered that the decay of the residual charge on a glass Leyden jar followed a stretched exponential law. The functional form that he discovered is often associated with a broad distribution of relaxation times, and has been found to describe a remarkably wide range of physical processes. To describe damping that is of the stretched exponential type, Kelvin chains or Maxwell elements in parallel have been used. Although an improved fit to the data can be realized by this means, the technique results in a high number of material parameters which have to be identified.
3.12
Fractional Calculus
A promising alternative to multi-exponential decay models is to replace classical rheological dashpots by 'fractional' elements. It is claimed that with only a few parameters, material behavior of many viscoelastic media can be described over large ranges of time and/or frequency [Hilfer, 2000]. It may also be possible with fractional derivatives to treat the discontinuities that are sometimes present in decay [ Asa 1996]. The disadvantage of fractional calculus is (i) the increasing computational/storage requirements, and (ii) the esoteric mathematics which is alien to the training of most.
3.13 Modified Coulomb Damping Model
Published here formally for the first time, with details described later; the heuristic 'modified Coulomb' model is an alternative to all of the aforementioned damping models. It is thought to be closely related to secondary creep [Peters, 2001, "Creep..] and (like fractional calculus) accomplishes good fits with a small number of parameters. Developed from energy considerations, its equations are expressed in canonical form involving the quality factor Q.
3.14 Relaxation
Formally, relaxation is defined by the behavior of a sample subjected to a constant strain. Because of the mechanisms just discussed in relationship to creep, the stress relaxes exponentially toward zero (in the simplest approximation). In practice, the definition just given can be misleading; since the word 'relaxation' is used to describe a host of processes in which some quantity decays exponentially in time-- for example, the relaxation of strain at constant stress in the Kelvin-Voigt model of viscoelasticity.
Some of the viscoelastic
models using dashpots and springs have been quite successful in the limited
regime of their applicability. For
example, the Zener (Debye)
model, which will be discussed again later, has been used for years to describe
a particular form of damping in solids, which derives from relaxations
associated with dislocations. Seminal
experimental work of this type was conducted by
Bordoni [Bordoni, 1954] performed experiments which led to his observation of relaxation-type internal friction processes where the acoustic attenuation is seen to peak at certain temperatures. The so-called Bordoni peaks occur at low temperatures or at ultrasonic frequencies. These losses, which are maximum when dislocation relaxations can take place in step with the driving frequency, were first observed in the FCC metals; lead, copper, aluminum and silver.
Dislocation damping as just described is
characterized by a temperature dependent relaxation that exhibits Arrhenius behavior.
By plotting the internal friction versus reciprocal temperature, one may
estimate the activation energy of the process responsible for the damping. The following quote from the introduction of
the
In a recent private communication, Prof. Granato has indicated the following: "Dislocations do follow the Zener (or Deybe) form fairly well for the damping, but not for the elastic modulus. This is because the response to a stress is given as a Fourier series. The higher order terms in the series have little effect on the damping, but a strong effect on the modulus at high frequencies. This makes the modulus fall off more slowly than with the reciprocal frequency."
4. Hysteresis--more details
Hysteresis and creep are common to many systems, such as electromechanical actuators, especially when used at high drive levels. Their transfer function is influenced by 'rate independent memory effects' . The state of the actuator depends not only on the present value of the input signal but also on the nature of their past amplitudes, especially the extremum values, but not on rates of the past [Visintin 1996] . This statement is in support of the author's secondary creep model of hysteretic damping, where the amplitude of the previous turning point determines the magnitude of the internal friction force for the half-cycle that follows. One of the most dramatic examples of a memory effect is the demonstration mentioned above by Gross in the 1950's, concerning a twisted wire.
Damping complexities derive from the defect structures that are found in real materials and which give rise to hysteresis, which in the Greek language means to ‘come late’. Although most everybody seems to appreciate magnetic hysteresis at some level; too few individuals (at least in physics) have been trained in the mechanisms of mechanical hysteresis responsible for damping. Dislocations, for example, are usually an 'add-on' chapter to a solid state physics text--even though they are known to be indispensible with regard to actual, as opposed to idealized properties of materials.
In the case of ferrous materials, the magnetization of a specimen lags behind the field generated by an electric current, to which the specimen responds. In the case of real springs that do not obey Hooke’s law F = -kx, the displacement x lags behind the spring’s restoring force F. It is convenient to express the resulting hysteresis in terms of ‘intrinsic’ variables instead of x and F. Thus the strain e (fractional change in the spring’s length if it were a wire in tension) lags the stress s (force per unit area). Usually in engineering practice the stress is reckoned with respect to the external force (negative of the spring F), so that the equivalent to Hooke’s law is s = E e, where E is an elastic modulus descriptive of the material from which the spring is fabricated. In the case of a straight wire, E would be Young’s modulus, but for coil springs E is determined primarily by the shear modulus. Some of the ways in which hysteresis can be represented for a freely decaying oscillator are shown in Fig. 3. The generalized coordinate q would be spring elongation for the force case shown, or it would be strain when the ordinate quantity is stress. The graph of velocity versus displacement is referred to as a phase-space plot. It is commonly used in describing chaotic systems and if ‘strobed’ at the frequency of the oscillator, becomes the Poincare’ section. Notice that the circulation is of opposite sign when using external force as opposed to spring force, in addition to the curves occupying different quadrants. It is important to recognize this difference, particularly when discussing negative damping, where the oscillation amplitude builds in time, as illustrated in the right hand part of the figure.
Figure 3.
Three different ways to represent hysteresis
damping for an oscillator in free-decay.
Cases of both positive and negative damping are illustrated.
Though
not very common in mechanical oscillators, it is possible to realize negative
damping. One example is that of an optically driven pendulum, because of the LiF crystals that were placed in its support structure
(containing high density of color centers produced by radiation) [Coy 1997]. An interesting feature of this pendulum was
its unwillingness to entrain to the driving laser. There are also examples of negative damping
from aerodynamics, such as flutter.
Since buildings and bridges can experience negative damping in
catastrophic manner (
Another example of hysteresis that is very much like negative damping (though not usually labeled as such) is to be found in a heat engine [Peters 2001, "The Stirling..]. The motion is not simple harmonic, rather the speed with which the hysteresis curve is traversed (in pressure vs volume) increases as the size of the hysteresis loop increases. A larger loop (greater work done by the gas) results in higher rpm of the engine as opposed to a larger amplitude of the motion at constant period. The gas pressure provides a force similar to the Hooke’s Law force of the spring in a mass/spring oscillator.
It is usually assumed that hysteresis loops are ‘smooth’, which is not necessarily true. For example, in the case of magnetic hysteresis, the ‘jerky’ parts known as the Barkhausen effect [Barkhausen 1919] are well known. The equivalent jerky behavior in metallic alloys is known as the Portevin LeChatelier effect [Portevin 1923]. Although we have historically avoided these cases that appear to be intractable in a mathematics sense (not obeying the fundamental theorem of calculus), their presence is undeniable testimony of the complex nature of hysteresis.
5. Damping Models
5.1 Viscous Damped Harmonic Oscillator
As first seen
by students in a textbook the equation of motion for a damped, driven harmonic
oscillator is likely as follows,
(1)
where m is mass, k is spring constant, c is a 'constant' of viscous damping, and F(t) is the external force driving the oscillator. It is convenient to work with a coefficient of performance, or quality factor Q, and rewrite equation (1) in canonical form as
(2)
For F(t) = 0 and an assumed solution, x = Aexp(pq), with q=w0t, the differential equation becomes algebraic (quadratic) in x(q) with the roots given by
(3)
Depending on the value of Q, the motion is either overdamped (non-oscillatory), critically damped, or underdamped. Here we restrict our attention to the last case corresponding to Q > ½, in which the square root term of equation (3) is imaginary. Moreover, we are mostly concerned with systems in which Q >> 1.
5.2 Definition of Q
The
quality factor Q is in general defined as 2p times the ratio of the
energy of the oscillator to the energy lost to friction per cycle. For viscous
damping (and hysteretic damping, later discussed) the Q is independent of the
amplitude of oscillation. For other types of damping, we will see that the Q is
not constant. In the case of the viscous
damped oscillator, 
where b
appears in the solution as an amplitude decay
'constant'. The parameter b is not
really constant, as discussed in PART II, Section 9.
(4)
Since x is real, we use the real part of equation (4) and employ Euler’s identity to obtain
(5)
where f is a constant determined by the initial conditions.
5.3 Damping 'Redshift'
It
is seen that the frequency of oscillation depends on the damping constant, b;
however, the fractional change Dw/wo is almost always negligibly
small. For example, the reduction in
frequency is only 1.4 % for
Q=3; which is close to critical damping of Q=0.5. At these small values of Q, the lifetime of a
freely decaying oscillator is so short that the frequency is ill-defined
because of the Heisenberg uncertainty principle. At larger Q's, where the
frequency is well defined, the shift is ignorable; i.e., at Q=100, the
fractional shift is only 1.3 x 10-5.
In the case of internal friction (hysteretic) damping, there is no redshift anyway, because the oscillator is isochronous
5.4 Driven System
When F(t) is not zero, but rather corresponds to harmonic drive at angular frequency w and amplitude A; the response involves the sum of equation (5) (transient) and a particular solution (steady state).
. (6)
The system resonates (amplitude a maximum) at
, and the variation of the amplitude with w at
steady state at any drive frequency w is given by
(7)
The resonance response curve described by equation (7) is called the Lorentzian. More frequently in physics, the term is used to describe pressure broadened linewidths [Milonni 1988]. As noted previously, Lorentz was never apparently content with the damping term, 2bdx/dt. In his PhD dissertation concerned with the damping of electron oscillators through electromagnetic radiation, he was not able to satisfactorily describe the damping from first principles. Though we might be tempted to say that this failure derived from his classical (pre-quantum mechanics) description of the problem, such a viewpoint is an oversimplification.
5.5 Damping Capacity
5.5.1
Viscous damping
The loss per cycle, called the damping capacity, is computed for the viscous damping case as follows (per unit mass)
(8)
where A is the amplitude of the oscillation. Because the total energy per unit mass is w2A2 /2, we see that Q = w/(2b).
5.5.2
Hysteretic damping,
linear approximation
The equation of motion in this case is given by 
where h is a
constant. The energy loss in one cycle
is given by
(9)
so that Q = mw2/h.
5.5.3
Hysteretic damping,
modified Coulomb model
The nonlinear equation of motion introduced in this chapter to describe hysteretic damping is as follows:
(10)
where, in the last expression, the subscript 'prev' implies amplitude at the last (previous) turning point of the motion. This particular form for the damping term [Peters 2002, ... universal..] thought to result from secondary as opposed to primary creep [Peters 2001,.."Creep..] is not as computationally useful as the middle expression involving the Q. The damping capacity is given by
(11)
yielding Q=pw2/(4c), so that the constant in the nonlinear model is related to the linear approximation constant through
c =p h /(4m).
5.6 Coulomb Damping
One of the simplest friction models
is that in which a Hooke’s law spring is connected on
one end to a mass that slides on a level table.
The other end of the spring is connected to a stationary wall. The
friction force of the mass against the table is of the type first described
quantitatively by Charles Augustin Coulomb
(1736-1806), although Leonardo da Vinci is probably
the first to consider it scientifically.
The equation of motion and its solution, for the free-decay of an
oscillator damped by Coulomb friction, is given by
(12)
The equation is nonlinear because of the sign of the velocity term, but it is easily integrated numerically; additionally, it is one of the few nonlinear equations for which an analytic solution is known and is given above (for more details, the reader is referred to [Peters 1997]. The integer, n, specifies the number of half-cycle turning points from t = 0, and Dx is the decrement (linear, not logarithmic, having units of m) per half-cycle. There are occasions to use equation (12)-- for example, problems in civil engineering where relative motion of members (slipping) occurs at a structural joint. The work against friction in one cycle can be obtained from energy considerations and is given by
(13)
which for small decrement yields
(14)
5.7 Thermoelastic damping
A microphone with Labview was used to analyze vibratory data of an aluminum rod. A rod of 1 m length can be excited to ear-piercing intensities by holding at its center between thumb and finger of one hand, and stroking along the length with the other hand that is coated with violin-bow rosin. The decay of this "singing rod" , which is a common part of physics demonstration equipment, was found to be in agreement with the following theoretical expression for thermoeleastic damping [Landau 1965].
(15)
where w is the vibrational angular frequency, T is the temperature, r is the density of the bar, C is the heat capacity per unit volume, a is its thermal expansion coefficient, and k is the thermal conductivity. The expression assumes adiabatic vibrations and there is no thermoelastic dissipation in pure shear oscillations (e.g., torsional oscillations of a bar), because the volume does not change and hence there is no local oscillation of the temperature. Notice in particular that the Q is inversely proportional to frequency, unlike viscous damping that is proportional to the frequency, or hysteretic damping that is proportional to the square of the frequency. Thermoelastic damping is important for high frequency compressional oscillations in materials with significant thermal coefficients and especially for metals, because of their large thermal conductivity.
The demonstration of comparable behavior in polymers (entropy spring, but opposite sign compared to metals) is quite easy. Stretch a rubber band between the hands and immediately touch it to the forehead. The increase in temperature is easily sensed. Conversely, releasing the tension in the band cools it enough to be sensed by placing the band to a part of the face that is sensitive to temperature change. Equation (15) does not apply to polymers.
6. Measurements of Damping
6.1 Sensor considerations
The
challenge to any measurement is to accomplish the task without significantly
altering the system under study. For
measurements on mechanical oscillators of the type described in this document,
two types of sensor are generally superior to every other kind; they are (i) optical, and (ii) capacitive. Optical sensors are probably the least perturbative but they do not readily yield themselves to
large dynamic range with good linearity (and small quantization errors for
digital type). Inductive sensors, such
as the LVDT, are known from seismology to be inherently more
noisy (up to a hundred times) because of ferromagnetic granularity. Additionally, transformers are not amenable
to miniaturization; and the components are inherently less stable. It is therefore a mystery--the widespread
use of the fully differential inductive sensor (LVDT) instead of the superior
fully differential capacitive sensor; which is electrically equivalent, apart
from its reactance type--and also capable of miniaturization to the NEMS
level. The challenge with really small
capacitive sensors is the increase in output reactance of the device as they
approach femtoFarad levels of individual
capacitors.
All
measurements reported in this document were taken with the fully differential
unit whose patent name is "symmetric differential capacitive" [c.f.
Peters, 1993: "Full-bridge...]. It
is especially useful for studying mechanical oscillators of macroscopic size;
and morphed to various forms, it recently has found application in
micro-electro-mechanical systems (MEMS). It is capable of great sensitivity
when configured in the form of an array, as shown in Fig. 4.
Figure
4.
Illustration of a fully-differential capacitive transducer
array. For clarity, the three electrode-sets are
shown separated from their operating positions (parallel with a small
separation gap, with the moving electrodes in the middle).
Various lines in Fig. 4 correspond to narrow insulator strips, such as the single vertical line in the set that connects to the amplifier. In the cross-connected static set, the plates labeled '1' are electrically distinct from the others labeled '2'. The total-plate arrangement constitutes a symmetric a.c. bridge, and the central position of the moving set (x = 0 as shown in the figure) corresponds to bridge balance with V0 = 0. Displacement away from balance gives a voltage output that is linear between -w/4 and w/4, as illustrated in the graph at the bottom of the figure.
The oscillator frequency is typically tens of kHz, and the amplifier is of instrumentation type [Horowitz 1989]. Unlike a bridge null detector, the linear response through x = 0 is realized when synchronous detection is employed. This can be accomplished with a lock-in amplifier, but the most recent Cavendish balance to employ the sensor uses diodes [Tel-Atomic Inc., online at http://www.telatomic.com/sdct1.html]. A tutorial ('detailed explanation') of the SDC sensor using diodes can also be found at this website.
In
Fig. 4, four individual SDC units have been shown connected in parallel. The
total number N of individual units in an array depends on the characteristic
width w, for which the total range of detectable motion is w/2. If the requirement on range
is small, then N can in principle be made very large, which is desirable for
the following reason. The
sensitivity of this position sensor is inversely proportional to w, if output
capacitance of the device were not a factor.
As w is reduced, however, the degrading influence of increased output
reactance (capacitive) is more significant than the improved sensitivity that
would result if the sensor could be connected to an amplifier with infinite
input impedance. Since the
instrumentation amplifier's input capacitance is not negligible, shrinking w is
beneficial only if the output reactance can by some means be kept low. This is accomplished with the array of
individual units. In principle, the
output capacitance could be held reasonably constant as N approaches 100, by
using photolithographic techniques and small spacing between the parallel
electrodes. The concept has been deemed feasible because of existing
technologies as well as the following: though not in the form of an array,
No doubt the popular silicon based MEMS
accelerometers marketed by Analog Devices take advantage of the impedance
advantages of an array, employing a large number of 'fingers' in a
force-feedback arrangement. Although employed
mostly otherwise, the first case of a fully differential capacitive transducer
using force-feedback was one based on simultaneous action of actuator and
sensor functions in a single unit of nonlinear type [Peters 1991].
6.2
Common-mode rejection
In attempts to measure damping, one can be confronted with difficulties of mode mixing. For example, the historical Cavendish experiment, using optical detection, has been traditionally difficult, unless the instrument is placed in a very quiet location to avoid pendulous swinging of the boom. The high-frequency pendulous motion (of the order of 1 Hz) as a 'noise', becomes superposed on the low frequency torsional signal. The computerized Cavendish balance sold by Tel-Atomic overcomes this problem by means of a mechanical common-mode rejection feature. An SDC sensor placed near one boom end is connected in electrical phase opposition to a second SDC sensor placed near the other end of the boom. The boom itself serves as the moving electrode for both sensors. Neither sensor has a first-order response to boom motion parallel to its long axis. Pendulous motion perpendicular to the boom orientation is largely canceled.
6.3 Example of viscous damping
The
aforementioned Sprengnether LaCoste
spring seismometer is well-suited to the demonstration of viscous damping, when
damping is imposed in the following manner:
the instrument was built with a Faraday law (velocity) detector; i.e., a
coil that moves with the mass of the instrument, in the field of a stationary
magnet. As originally employed, the coil
was connected to the amplifier of a recorder.
In the present configuration, however, the velocity detector is not
employed; since its sensitivity is severely limited at low frequencies. Instead, an SDC array of the type shown in
Fig. 4 is used to measure the position of the 'mass' (a pair of lead weights,
total mass 11 kg). If the instrument is
operated with the coil open-circuit, there is no induced current. By connecting a resistor across the coil (through
very fine copper wires that go to terminals on the case) mass motion induces a
current. The induced current opposes the
motion through Lenz's law, resulting in damping. The damping depends on the size of the
current and is thus an inverse function of the resistor's magnitude through
Ohm's law. The phenomenon is illlustrated in Fig. 5.
Figure 5. Examples of induced current damping of a vertical seismometer, using two different resistors
As compared to the 'undamped' instrument, whose Q is approximately 80 at a period of 17 s, it is seen that the addition of a 990 ohm resistor lowered the Q by more than an order of magnitude to 4.9. A 330 ohm resistor reduced it even further to 3.1. The amount of damping is governed also by the resistance of the coil winding, which is 480 ohms.
The envelopes that have been fitted to the decay curves were the basis for estimating the Q. The decay data were imported to Excel, by first outputting the Dataq DI-154RS A/D generated record as a *.dat (CSV) file. The fits were produced by trial and error using the 'drag' and 'autofill' function. Notice that the 990 ohm resistor (first) case is not as pure an exponential decay as the other case because of creep. The rate of creep is larger at large initial amplitudes of the motion.
6.4 Another way to measure damping
Curve fitting (full nonlinear in general) is the best way to estimate damping parameters, especially if the decay is not exponential. For more routine cases, simpler methods can be used. Among the host of ways that have been defined to specify the damping of an oscillator, one of the most common uses the logarithmic decrement. The solution to equation (1) with zero right hand side is given by
. (16)
The
full-cycle turning points, 
, with N = 0, 1, 2,
…can be used to compute the logarithmic decrement through
(17)
Unfortunately, an estimate based on equation (17) can be difficult due to the presence of either or both of two problems: (i) mean position offset in the decay record, or (ii) asymmetry of the decay, where the turning points on one side of equilibrium decay at a different rate than those of the other side. Case (ii) occurs more frequently than one might expect; it is frequently a consequence of material complexity and not the result of nonlinearity in the electronics of the detector. It is important, however, to be sure that the detector is either linear or that corrections for the nonlinearity be utilized before estimating the damping.
A method to provide partial compensation uses half-cycle turning points n = 2N, and works with a minimum of three such points.
(18)
Advantage is taken of random error reduction by using equation (18) on a set of turning points (optimal number sometimes being about a dozen). The calculations are straightforward in a spreadsheet such as Excel, by means of the ‘autofill’ function.
7. Hysteretic Damping
7.1
Equivalent viscous
(linear) model
The few mechanical oscillators governed by equation (1) tend to be those in which there is an external control, such as eddy current damping. For oscillators in which the damping derives from internal friction of its members, the following linear approximate form of the hysteretic damping model has been used
(19)
It
should be noted that hysteresis is the cause for all
damping, however engineers have come to use the term, hysteretic damping, for
systems described by equation (19). This
equation differs in two important ways from equation (1). For the viscous damped oscillator Q is
proportional to the frequency, but for the hysteretic damped oscillator, Q is
proportional to the square of the frequency.
Also, viscous damping
changes the frequency of the oscillator, since w1
< w0 ; and for resonance the frequency is even lower. But the hysteretic oscillator is isochronous,
requiring only a single frequency 
to describe all
features of the motion. For example, it
is easy to show that the oscillator resonates at this frequency. Off resonance, the response is not the
standard Lorentzian.
To show this, assume steady state and use the phasor
method given to us by Steinmetz (complex exponential form for the variables);
i.e., F = F0 ejwt and x = x0ejwt to get
the frequency transfer function
(20)
for which the real and imaginary parts are given by
(21)
which is expressible in polar form as
(22)
It
is interesting to compare the steady state response of the driven, hysteretic
damped oscillator to that of the driven, viscous damped oscillator; i.e., equation (22)
compared with normalized equation (7). A
Bode plot comparison (log-log, for the amplitude case) is provided in Fig.
6. At small values of the damping
parameter a
(large Q) there is insignificant difference between 
Figure 6. Bode plot comparison of steady state driven system with (i) hysteretic damping (dark curves) and (ii) viscous damping (light curves).
the two cases. At large values, however, the difference is significant.
7.2 Examples from experiment of hysteretic damping
The vertical seismometer that was used for several of the present studies is known to decay according to hysteretic damping. In section 16.4 titled "Failure of viscoelasticity", are provided details of the work by Gunar Streckeisen that showed this to be true. Decay curves of the instrument are frequently a near perfect exponential, once corrected for secular drift of the record. Sometimes this drift is the result of creep in the spring of the instrument, but it may also be the result of other factors, such as (i) temperature change or (ii) barometric pressure variation, or even (iii) tidal influence. The temperature sensitivity is due to the difference of thermal coefficients of the materials from which the instrument is constructed, and the pressure variation is a buoyancy effect. Tidal influence is the smallest of the three, which causes minute accelerations of the crust of the earth with a period of about 12 hours.
In the discussions which follow, two different decay records are provided. In both cases the initial amplitude of oscillation is quite large, being a significant fraction of 1 mm, and the period for the two cases is different--the first case being 17 s and the second one 21 s. The first case time record, shown in Fig. 7, contains 9800 points. Once a 12 mV/s (upward) drift was removed, the decay (left curve) is seen to be 'nearly textbook' exponential.
Figure 7. Free-decay of
a vertical seismometer due to hysteretic damping. The period of oscillation is 17 s.
The adjective 'nearly' is appropriate, because there is a 12% difference in the decay constants defining the upper and lower turning points (0.0022 top, 0.0025 bottom), which were determined by trial and error 'eyeball' exponential fits using Excel. In this author's experience, such is the norm for virtually all mechanical oscillations; perfectly symmetric exponential decays have rarely been seen in the hundreds of cases studied.
Because there are roughly 150 oscillations in the time record of Fig. 7, it is not possible to resolve individual turning points of the motion, but the oscillations are very nearly that of a pure damped sinusoid, as noted from the right-side graph of the figure. This spectrum was generated with a 4096 point FFT, comprising the first 1090 s of the time record. The 2nd harmonic is the only distortion observed, and it is about 65 dB below the fundamental. For the case presented in Fig. 7, Q = 80.
Another example of free-decay hysteretic damping
is provided in Fig. 8. As usual, the
record was afflicted by drift, possibly from creep in the spring; in this case
a constant rate of 60
microvolts per second, as observed in the graph on
the right. All of these graphs were
produced with Excel, as noted earlier in the discussion of induced current
damping. As with the decay curve of Fig.
7, the creep-corrected graph on the left was generated by adding a secular term
to the raw data. Once corrected, the
decay is a near perfect exponential of hysteretic type. We will see other examples (from pendulum
studies) in which 
Figure 8. Free-decay of
a seismometer due to hysteretic damping.
two damping mechanisms are simultaneously active in a decay.
The Q values corresponding to figures 7 and 8 are consistent with hysteretic damping; i.e, 80 for the 17 s oscillation and 52 for the 21 s oscillation. As noted elsewhere in this document, Q a w2 for hysteretic damping as opposed to an exponent of 1 for viscous damping. Of course, one must collect data over a very much larger range of frequencies to verify this, as was done by Streckeisen.
8. Failure of the Common Theory
Many mechanical oscillator studies in decades past, mainly by engineers, have shown that the so-called decay constant b is proportional to w-1 instead of being constant [c.f. Bert 1973]. The damping for these cases came to be called "material", "structural", or "hysteretic". A common means to obtain the correct frequency dependence was to divide the velocity by frequency and call the result an "equivalent viscous" form of damping. The adjective, "equivalent", draws attention to the fact that internal friction in a solid can't really result from fluid effects. Moreover, elsewhere in this document there is plenty of support for the position that the linear equations of viscous damping type cannot produce truly meaningful (predictive) models when doing modal analysis on many-body systems.
An important early work by Kimball and Lovell [Kimball 1927] is evidently the first experiment to show that internal friction ('force') of many solids is virtually independent of frequency. In other words, their elegant technique in which a rotating rod is deflected by a transverse force, was the first to demonstrate the 'universality' of hysteretic damping. Although both researchers were physicists at General Electric in the time of Steinmetz; few physicists of the 21st century know of this important work. As with the important contributions of Portevin and LeChatelier, their study of systems influenced by 'dirty physics' was evidently ignored in favor of the 'clean' new quantum mechanics of that era.
It is interesting that a bell made of lead does not tinkle at room temperature, but it can be made to do so at 77 K, by immersion in liquid nitrogen. This demonstration, which is often employed in physics 'circuses', shows clearly that the internal friction of lead at audio frequencies can be reduced substantially by lowering the temperature. An important lesson to be learned from these observations is that damping, in general, is a complex function of temperature, frequency, conductivity, ..... (who knows where to terminate this list). Not only are a multitude of state variables necessary for a complete description of dissipation, but the previous history of stress-strain cycling may also be critical. Such is the nature of defect structures responsible for damping.
9.
Air Influence
Even
when operating an oscillator in high vacuum, there is a significant remanent damping that derives from internal friction. This
fact is illustrated in Fig. 9, which provides data for two different 'simple' pendula. They are simple in the sense that the bob
mass is concentrated near the bottom of the pendulum structure. In the figure,
decay time (reciprocal of the decay constant, b ) has
been plotted against the natural log of the pressure in mTorr.
Pressure reduction was done with a high quality roughing pump, and the pressure
was measured with (i) a mechanical gauge in the range
8 Torr < P < 760 Torr
and (ii) a thermocouple vacuum gauge for 0 < P < 100 mTorr.
In the range from 100 mTorr to 8 Torr,
the pressure could not be accurately measured with either of these gauges.
Similarly, pressures below 1 mTorr could not be
presently measured, but in similar other experiments with this pump, and using
an ion gauge; it was easy to pump below 0.01 mTorr
The period of each pendulum was very close to 1 s, and the starting amplitude of the motion for every case was about 25 mrad. The heavier pendulum used a pair of pointed steel supports resting on single-crystalline silicon wafers to provide the axis of rotation. At the bottom of the pendulum was attached a solid lead ball whose mass was approximately 1 kg. The lighter pendulum was supported by a steel knife edge resting on hard ceramic flats; and a large (10.3 cm dia.) lightweight (143 g) hollow metal sphere was attached at the bottom, to provide as much air drag as possible. The motion was measured with an SDC sensor feeding the computer through a Dataq DI-154RS analog to digital converter
Figure 9. Pendulum damping as a function of pressure in a vacuum chamber.
Although
air damping is evident in Fig. 9, it is not as influential as one might expect,
at least for the heavy pendulum. Moreover, at atmospheric pressure it was easy
to demonstrate the importance of nonlinear drag. As also noted in [Nelson 1986], this form of
fluid friction caused a significant amplitude
dependent damping.
The remanent damping, once air influence was eliminated (pressure below 1 mTorr) is substantial, relative to atmospheric damping, for both pendula. Removing the air increased the Q from 7500 to 10,100 for the heavy pendulum, and for the light pendulum, the increase was from 1000 to 4600. We thus see that even a pendulum designed to be heavily influenced by air drag also has significant damping that depends on the material from which the pendulum is fabricated, or on the material upon which it rests.
The difference in internal friction damping between the heavy and light instruments was not expected to be so great. Although this might be due to the difference in axis-type (points for the heavy instrument and knife-edge for the light), no systematic effort was made to determine the primary source of the damping difference. In addition to different axis designs, the means for holding the instruments together was different. The light pendulum used a large diameter solid brass wire between the axis and the lower mass, and the heavy pendulum used an aluminum tube.
Both of the pendula
used to generate Fig. 9 were relatively
high-frequency instruments (period of 1 s). The pivot was located, in each
case, near the top end of the instrument. As such, they stand in stark contrast
with the instruments that motivated this paper--where long-period pendula were used. A simple instrument to demonstrate some
of the complexities of long-period instruments is a rod-pendulum of adjustable
period [refer to Fig. 1 above]. The closer the axis to the
center, the longer the period and the greater the influence of internal
friction. It is easy to show that the sensitivity of a pendulum to
external forces is proportional to the square of the period. Similarly, the
ability to detect influence of internal configurational
change is quadratic in the period.
10. Noise and Damping
10.1 General considerations
Damping is inseparable from noise issues having nothing to do with undesirable sounds that might be produced by oscillation. In the simplest cases, the noise associated with damping can be described by the fluctuation-dissipation theorem. The viscous damped, thermally driven oscillator is a classic example of thermodynamic equilibium, for which this theorem is applicable. The classic electronics analogous case is the Johnson noise of a resistor, described by the Nyquist (white) noise formula.
The largest obstacle to constructing a perfectly simple harmonic oscillator is the oscillator's dissipation. If damping were perfectly smooth, this would not be so great a challenge. But the fluctuation-dissipation theorem of statistical mechanics guarantees that damping is accompanied by fluctuating forces. The larger the damping, the larger the fluctuating forces; i.e., the larger the noise. It is a standard problem in statistical mechanics to show that the magnitude of relative fluctuation is inversely proportional to the square root of the number of particles involved. In the case of internal friction noise, defects associated with mesoscale structures cause the effective number of particles responsible for the noise to be much smaller than the total number of atoms in a sample. Unfortunately, the fluctuation-dissipation theorem probably does not apply. It has been long known that it does not apply to the Barkhausen effect [Barkhausen 1919]. It has been recently demonstrated that it does not apply to structural glass [Grigera 1999]. The close relationship postulated by the author between the PLC effect and the Barkhausen effect implies that the fluctuation-dissipation theorem should also not generally apply to internal friction damping.
Internal friction noise is not white but rather
more like 1/f (or flicker = pink) noise, a ubiquitous form that has not
yet been explained from first principles.
A frequently cited paper on self-organized criticality states the
following: "We shall see that the
dynamics of a critical state has a specific temporal fingerprint, namely
'flicker noise', in which the power spectrum S(f)
scales as 1/f at low frequencies. Flicker noise is characterized by
correlations extended over a wide range of timescales, a clear indication of
some sort of cooperative effect. Flicker noise has been observed, for example, in the
light from quasars, the intensity of sunspots, the current through resistors,
the sand flow in an hourglass, the flow of rivers such as the
It should be noted that controversy exists concerning this self organized criticality paper, summarized in the following excerpt from Bak's book on 1/f noise called How Nature Works: The Science of Self-Organized Criticality (page 95): "In an earlier work ([CFJ]), performed while an undergraduate student in Aarhus, Denmark, [Kim Christensen] showed that our analysis of 1/f noise in the original sandpile article was not fully correct. Fortunately, we have since been able to recover from that fiasco in a joint project by showing that for a large class of models, 1/f noise does indeed emerge in the SOC state."
In the last few years
mathematicians have been drawn to "...an analogy, in which three areas of
mathematics and physics, usually regarded as separate, are intimately
connected. The analogy is tentative and
tantalizing, but nevertheless fruitful.
The three areas are eigenvalue asymptotics in wave physics, dynamical chaos, and prime
number theory" [M.
10.2
Example of mechanical
1/f noise
Shown in Fig. 10 is an example of mechanical flicker noise made worse by creep, that originates in the spring (LaCoste type) of a Sprengnether vertical seismometer. The data are from two separate time records, the 1st run preceding the 2nd run by about a half-hour. Just before collecting the data of the 1st run, a clamping pin was removed from the seismometer. Used to constrain the mass from moving, this pin had been left in place overnight to determine the amount of electronics noise, including drift. The measured electronics noise (white = 1/f 0 ) was more than an order of magnitude smaller than the smallest (high frequency) noise components of mechanical (seismometer) type. The peak-to-peak amplitude of the oscillation in both cases was 0.5 mm (calibration constant for the sensor being 2000 V/m). The peak to peak amplitude for SNR =1 for this system is of the order of 1 mm.
Figure 10. Evidence in support of 1/f mechanical noise in a seismometer.
Although the spring force was not unloaded with the pin in place overnight, nevertheless its removal caused a significant change to defect structures in the spring, as noted by the residuals between the data and their harmonic fits (magnified by a factor of ten).
The flicker character of the noise was demonstrated by computing power spectra of the residuals (not shown). The log-log plot, generated with the fast Fourier transform (Cooley Tukey FFT), showed 1/f frequency dependence for the second run. The larger noise of the first run was concentrated in the upper frequencies. The relaxation toward 1/f character sugggests that flicker noise is a remanent of 'work hardening' . To demonstrate that the flicker noise was not due to the electronics, sensor output was recorded with the mass of the instrument locked. The electronics noise proved to be more than an order of magnitude smaller and 'white' in character, probably mainly the result of A/D quantization.
Because the spring was not in equilibrium at the time the pin was removed, perhaps because of temperature change while the system was clamped; a great deal of initial molecular rearrangement occurred, involving atoms at grain boundaries. It is seen that the amount of fluctuations has noticeably decreased during the half-hour separating the two runs. Although the creep-noise would be undoubtedly much greater if the spring were relaxed altogether, it is not easy in such a case to quickly rebalance the seismometer to oscillate with a period in excess of 20 s. These observations are in keeping with known properties of sensitive seismometers as noted by Erhardt Wieland in "Instrumental self noise--transient disturbances" [ed. Borman 2002]: "Most new seismometers produce spontaneous transient disturbances, quasi miniature earthquakes caused by stresses in the mechanical components. Although they do not necessarily originate in the spring, their waveform at the output seems to indicate a sudden and permanent (step-like) change in the spring force. Long-period seismic records are sometimes severely degraded by such disturbances. The transients often die out within some months or years; if not, and especially when their frequency increases, corrosion must be suspected. Manufacturers try to mitigate the problem with a low-stress design and by aging the components or the finished seismometer (by extended storage, vibrations, or alternate heating and cooling cycles). It is sometimes possible to relieve internal stresses by hitting the pier around the seismometer with a hammer, a procedure that is recommended in each new installation." [ Wielandt 2002].
Material damping noise appears to have features that are similar to Barkhausen noise--a magnetic phenomenon involving a system far-from-equilibrium. Such noise is associated with the granular nature of ferromagnetic domains and has consequence in the design of electronic instruments using iron alloys. For example, it is known (though not widely) that the popular linear variable differential transformer (LVDT) is inherently less sensitive than a capacitive sensor of equivalent electrical type, because of its ferrous component (the rod-component that moves). As noted by Wielandt, the capacitive sensor ".... can be a hundred times better than that of the inductive type" [ Wielandt 2002]. Fully differential capacitive sensors, being electrically equivalent to the LVDT have still greater advantages borne of the higher symmetry. Additionally, by configuring the capacitive device as an array, it is possible for the sensitivity to also be greater.
Barkhausen noise and
hysteretic damping noise may be much more similar than has been
realized--involving granularity at the mesoscale,
intermediate between micro and macro-phenomena. For such systems, first principle methods
are very difficult to employ, due to complexities that originate from a host of
nonlinear interactions. For example, in
the case of internal friction of solids, damping derives from stress-strain hysteresis determined by defect structures in the
solid. Involving roughly 1012
atoms per 'grain' in metal specimens, flicker noise
evidently derives from self-similar structures of fractal geometry. It involves a higher degree of spatial
correlation than is true of white noise. The ubiquitousness
of 1/f noise is consistent with the labeling of hysteretic damping as
'universal', as first suggested by Kimball and Lovell.
10.3 Phase noise
The previous exmple was concerned with amplitude noise. It is also possible to see phase noise of mechanical type, as illustrated in Fig. 11.
Figure 11. Illustration of phase
noise in the free-decay of a vertical seismometer.
For the sensor calibration constant of 2000 V/m, it is seen that the initial amplitude of oscillation is 1.3 mrad, which is much too large to observe the discontinuities of mechanical Barkhausen type. The phase noise is made obvious by comparing the decay to a 'reference'; i.e., by looking at the residuals from an Excel generated fit to the data. By adjusting in a computer-generated damped sinusoid (i) the initial values of amplitude and phase, (ii) the decay parameter, and (iii) the frequency; one can visually by trial and error come close to an optimum fit to the data. Having done so with Fig. 11, the striking feature of the residuals (difference between data and fit) is the structure that looks something like 'beats' but is not. The phase noise responsible for this behavior is thought to be consistent with 1/f mechanical noise. To visualize the noise, it is convenient to think in terms of a small randomizing (noise) vector whose tail is positioned at the head of the phasor used to generate the record. The component of the noise vector that is in the direction of the the phasor generates amplitude noise as in Fig. 10, whereas the perpendicular component is responsible for the phase noise of Fig. 11.
Vibration
phase noise imposes a serious limit on the performance of precision quartz
crystal oscillators, since they are sensitive to acceleration. The phase noise
in these oscillators can be observed by beating against a reference oscillator
of known character; i.e., the reference serves the same purpose as the computer
'fit' of Fig. 11. To reduce the phase noise, crystals are isolated with low
natural frequency vibration isolators [as described in the marketing literature
of Wenzel Assoc.,
11. Transform Mathematics
11.1 General considerations
For linear systems, the Laplace
and Fourier transforms (
Ideas concerning the fast Fourier transform (FFT) were evidently originally treated by Gauss in the early 1800's, but the digital signal processing (DSP) 'explosion' of the 1960's was largely due to the work of Cooley and Tukey [Cooley 1965]. For an interesting historical account about an "accident" in the publication of their paper, the reader is referred to [Cipra 1993]. There, Cipra says the following about the FFT: "The Fourier transform stands at the center of signal processing, which encompasses everything from satellite communications to medical imaging, from acoustics to spectroscopy. Fourier analysis, in the guise of X-ray crystallography, was essential to Watson and Crick's discovery of the double helix, and it continues to be important for the study of protein and viral structures. The Fourier transform is a fundamental tool, both theoretically and computationally, in the solution of partial differential equations. As such, it's at the heart of mathematical physics, from Fourier's analytic theory of heat to the most modern treatments of quantum mechanics. Any kind of wave phenomenon, be it seismic, tidal, or electromagnetic, is a candidate for Fourier analysis. Many statistical processes, such as the removal of "noise" from data and computing correlations, are also based on working with Fourier transforms.".
Concerning the last statement about noise, this author has used autocorrelation as a powerful means for identifying short-lived, low frequency periodic signals in time records, that do not readily show up in power spectra (FFT's). For example, they are the means for studying free-earth oscillations--eigenmodes excited by rapid relaxations of the Earth under tidal stressing (12 hour periodic).
[Peters 2000]. The FFT is used to generate the autocorrelation by means of the Wiener-Khintchine theorem [Press 1986] .
The great advantage of the FFT compared to the DFT has to do with degeneracy. The DFT proceeds to calculate the components of every 'vector' in the reciprocal space (frequency reciprocal to time, units of 'second'; or wave number (spatial frequency) reciprocal to displacement, units of 'meter') with disregard for the fact that many components have the same value, apart from a change of sign.
11.2 Bit reversal
The key to the power of the FFT (CPU time proportional to n log n) compared to the DFT (CPU time proportional to n2 ) is the bit reversal scheme of the Cooley Tukey algorithm. It is illustrated very simply by the following: Instead of a practically-sized number of samples in the record to be transformed (minimum of n = 1024, typically), consider (for pedagogy) n = 8, distributed on the unit circle as shown in Fig. 12.
Figure 12. Graphical illustration of why the Cooley Tukey FFT algorithm is much, much faster than the outdated
DFT.
Observe that the roots of unity in the complex plane, which have been numbered 0 through 7, divide the 'pie' into 8 equal pieces. (The algorithm requires that n be expressible as a power of 2). The usual decimal counting scheme for the 8 'vectors' is as indicated, traversing the phasor diagram (circle on left) sequentially. In the Cooley-Tukey algorithm, a choice is made to reverse the bits of the binary representation of the vector. Usually, the least significant bit is on the right and the most significant bit on the left; so that decimal counting is as shown on the right in the table--from 0 to 7. With bit reversal, 'lsb' becomes the left most binary digit and the 'msb' is the right most digit. Thus, for example binary 110 (usually 6) becomes 3.
Using bit reversal, the phasor diagram is not traversed in the usual phasor (circulatory) sense, but rather in a 'flip-flop' back and forth across the circle. By this means, there is no needless repetition in the calculation of 'vector' components (real and imaginary values of a given term in the transform). For example, 5 is the simple negative of 1. It is much faster to reverse the sign on 1 to get 5, than to needlessly calculate values for sine and cosine terms a second time. The savings in time is substantial as n gets large, since there are then a great number of circulations of the phasor circle. For a 1K record, the FFT computes the transform 102.4 times faster than does the DFT. Additional details are provided in [Peters, 2003, ....speed...]
11.3 Wavelet transform
Recent work suggests that the wavelet transform (WT) may in the future replace the Fourier transform in some applications. It uses the Haar function, which is orthogonal on [0,1], as opposed to the orthogonality of the harmonic functions (sine and cosine) corresponding to [0, 2p]. [Strang 1993]. It is claimed that the WT is better able to address features of the Heisenberg uncertainty principle than the FFT.
11.4 Heisenberg's famous principle
The heart and soul of quantum mechanics is the
Heisenberg uncertainty principle. As
noted elsewhere in this document, it has things to say about damping
models. According to well known
physicist Hans Bethe, the principle has received 'bad
press': "Many people believe that the
uncertainty principle has made everything uncertain. It's quite the opposite.
Without the uncertainty principle there could not exist
any atoms, there could not be any certainty in the behavior of matter. So it is
in fact a certainty principle."
Curiously,
a failure figured into Heisenberg's discovery of the principle. During his
thesis defense, in front of great theoretical physicist Arnold Sommerfeld (his director) and the famous experimentalist
Wilhelm Wien, he proved unable to derive the
magnifying power of a simple microscope. The scandal culminated with Professor Wien asking him to explain how a battery works; and he
could not answer that question either.
Knowing his extraordinary theoretical giftings,
Sommerfeld gave him the highest possible grade to
compensate for Wien's choice of an F. Thus Heisenberg
was awarded his doctorate.
Later,
in an ironic turn of events , Heisenberg chose a
microscope to illustrate features of the matrix quantum mechanics that he
originated; and which corrected problems with the Bohr wave mechanics
theory. His greatest source of
embarrassment served to make Heisenberg famous!
12. Hysteretic damping
12.1 Physical Basis
The model of simple harmonic oscillation with
viscous damping assumes dissipation from an externally acting force. It is not suited to a conceptual understanding
of hysteretic damping. To accommodate
internal friction requires more than a single mass connected to the elastic
component responsible for restoration.
Two systems are pedagogically useful in this regard, one being a
long-period physical pendulum (mechanical), and the other being the oscillator
used by Ruchhardt to measure the ratio of heat
capacities of a gas (mainly thermodynamic).
Because of widespread confusion concerning the difference between
viscous and hysteretic damping, both cases are presented here. The treatments are provided as evidence for
the premise that hysteretic damping is the more important case for applied
physics and engineering.
It is common knowledge that the damping of a mechanical oscillator results from the conversion of mechanical energy into thermal energy. One might expect then, that a direct consideration of thermodynamics could yield conceptual understanding of the underlying physics. Although an ideal gas is rarely considered in this context, there is a classic experiment which speaks to its relevance. It is the ingenious technique used first in 1929 by Ruchhardt to measure g, the ratio of heat capacity at constant pressure to that at constant volume [described in Zemansky 1957] .
12.2 Ruchhardt’s Experiment
Consider a piston of mass m moving in a cylinder of cross-sectional area A, alternately compressing and expanding a volume of ideal gas V0, about the residual pressure P0. Assume that there is no sliding friction between the piston and the cylinder. A small displacement x of the mass results in volume change DV = V - V 0 = Ax. There is a restoring force F=ADP, where the pressure difference DP relates to DV through an assumed adiabatic process; i.e., the period of the motion is assumed too short for appreciable heat transfer into and out of the gas. Using PVg = constant, one obtains
(23)
from which
(24)
This
is the equation of motion of a simple harmonic oscillator. There is no damping because of the assumed
adiabatic process. By measuring the period 
one can estimate g.
Historically, it appears that such measurements slightly underestimate g , which can be understood as follows.
The ideal gas equation of state PV=NkT yields, through differentiation
(25)
Notice the difference between equations (24) and (25). In (25) damping is possible (a type of 'negative drive' term) from temperature variations associated with heat transfer during traversal of the cycle. If it were possible for the oscillation to be isothermal (DT=0 at very low frequency, essentially quasistatic), then the frequency would be lower than that of the adiabatic case, since g >1 is missing from equation (25). In the isothermal case there would also be no damping, since the heat into the gas during compression would be balanced by that which leaves during expansion. The only way to get damping is for the paths of compression and expansion in a plot of pressure versus volume to separate; i.e., for there to be hysteresis. Reality must correspond to something between the two extremes of adiabatic and isothermal, with experiment obviously favoring adiabatic. The process must depart somewhat from adiabatic, however, since there is damping, which equation (25) shows to derive from temperature variations yielding hysteresis. It is interesting to look at the temperature variations relative to a ‘driving force’, Fd ‘(t) . In the Ruchhardt experiment, there must be small variations DT’(t) that lag behind x(t). (These are not the reversible temperature variations of the adiabat, onto which the DT’(t) are superposed.) By comparing with equation (25), the right hand zero of equation (24) may be replaced with a damping force that can be written in terms of the velocity as
(26)
where c = constant. Notice that the multiplier on the velocity is not simply a constant, but rather a constant divided by the angular frequency. The use of velocity is mathematically convenient, but the magnitude of the velocity (speed) is not expected to be a first order influence on the temperature changes of hysteresis type. The derivative of x with respect to time not only shifts the phase by 90 degrees, which accommodates the lag with which heat is transferred; but it also introduces a frequency multiplier through the chain rule. Thus to make damping proportional to the velocity would cause increased dissipative heat flow and thus increased damping as the frequency is increased. Since this does not happen, and lest we introduce a non-physical term into the equation, it is necessary to divide by the frequency. Replacing the right hand side zero of equation (24) with equation (26) we obtain the modified equation of motion, with damping
(27)
Additional justification for the form of the damping term in equation (27) can be realized by looking at cases where there is negative damping; i.e., c < 0. Such is true when the gas is caused to cycle as an engine. An illustrative case-study was that of a low temperature Stirling engine [Peters 2002, Stirling...], in which reasonable agreement between theory and experiment was realized through the use of an equation based on the same arguments used to derive equation (27).
It is seen that a straightforward modelling of Ruchhardt’s experiment to include damping yields an equation of motion that is in the form of hysteretic damping. It appears that for many systems in which the dissipation is dominated by internal friction, hysteretic damping is a near universal form.
12.3 Physical Pendulum
In the paper by Speake
et al [Speake 1999] one finds the following statement,
"the logarithmic decrement (Q-1) varied as the inverse of the
square of the frequency. We interpreted this as evidence that, in Cu-Be over
this frequency range, the imaginary component of Young's modulus was
independent of frequency, contrary to that which was predicted by the Maxwell
model." To fit their theory with experiment, they used a
"modified" Maxwell model with a distribution of time constants that
ranged from 30 s to more than 4000 s. Motivation for their continued modelling efforts derived partly from the observation by
Kuroda [Kuroda 1995] that anelasticity was cause for
some of the huge errors that have been present in estimates of the Newtonian
gravitational constant, G, by the time of swing method..
Although it gives
agreement with their particular experiment, the model of Speake
et al does not have the blessing of Occam's razor.
Moreover, their claim that damping derived primarily from their flex pivot of
Cu-Be may not be true. Other studies suggest that the material defining
the axis of a long period pendulum is for many cases no more important
(and sometimes much less important) to the damping than the
material from which the pendulum proper is constructed. A model which
also agrees with experiment of the type they conducted, but which is simpler,
is now presented.
Illustrated in
Fig. 13 is an idealized long-period mechanical oscillator which could be
labeled a `physical pendulum'. The top and bottom masses are the same M,
assumed to be much greater than the mass of the connecting structure, which is
represented by the curved line.
A primary mechanism for internal friction damping can be understood by looking at the external forces acting on the pendulum, which are pictured in the `negative displacement' (b) case. The upward 'normal' force that acts through the pivot (usually a knife-edge) is opposed by the pair of bob-weights situated left and right respectively of the axis of rotation. As the pendulum swings alternately between positive and negative displacements, the structure undergoes periodic flexure. It should be pointed out that internal friction could still be operative throughout the structure even without net bending; i.e., there could be complementary pieces of the structure undergoing compression and tension respectively. Even if the oscillator were in the weightlessness of space, a drive torque would result in dynamic reactionary forces that give rise to damping by this means.
Figure 13.
Idealized physical pendulum used to develop the modified Coulomb damping
model
Assume
that the masses are separated a distance 2L and the axis of rotation is DL above
the geometric center. Applying
(28)
(Note: Eq. 28 can be rewritten to accommodate larger displacements, where elastic nonlinearity gives rise to unusual behavior. The amplitude trend of the period is opposite to that of the gravitational nonlinearity--thus providing for improved isochronism. For details refer to [Peters, 2003, "Flex-pendulum...] ).
The difference in displacement of the masses involves an elastic term proportional to q1 and a dissipative term that depends on its time rate of change; i.e.,
(29)
where c is a dimensionless constant. This result can be obtained by the complex exponential Steinmetz (phasor) method. The equation is consistent with the common assumption that stress and strain are related through a complex constant. The angle d is the phase angle with which h ('strain') lags behind q1 ('stress'). To describe the motion of the lower mass, we can ignore the elastic part of h, since it does not contribute to the damping (or if the rod does not bend, assuming there still is damping as noted previously). We thus remove the subscript, and after some algebra obtain the result
(30)
which can also, in terms of Q=2p E/(-DE), be expressed as
(31)
If as a material property, d is independent of frequency, then Q is quadratic in the frequency; i.e., the damping of the pendulum due to internal friction is inversely proportional to the square of the frequency--even though the internal friction (determined by d) is itself frequency independent. It is important to note that the frequency dependence of internal friction is not to be equated with the frequency dependence of the Q of the oscillator--even though internal friction is frequently stated as simply 1/Q. This will be discussed in greater detail in section 13.2.
12.3.2 Test of Q dependencies
The dependence of Q on frequency and length in equation (31) was tested experimentally with a physical pendulum. A pair of Pb spheres, each of mass approximately 1 kg was each drilled through a diameter to pass the shaft of an aluminum alloy arrow (length approx. 70 cm) of the type commonly used with compound hunting bows. A second hole was drilled perpendicular to the first and tapped for a set screw. The shaft was sawed into two pieces which were rigidly rejoined around a carbon-steel knife edge using force fit and epoxy to machined protruberances above and below the knife-edge(s). The knife-edges extend perpendicularly outward on opposite sides of the arrow at its center.
12.3.3 Simple method to measure damping
Although an SDC sensor could have been employed
instead, the experiments to be described were performed with a measurement
technique that warrants description because of its novel simplicity -- yet it
is reasonably accurate. To measure both period and damping, a small 'flag' was super-glued to
the top of the upper shaft. This flag
was a small, thin 'U'
shaped piece of plastic in which the upper legs of the U were about 1 mm wide,
with a spacing between centers of about 0.5 cm.
An infra-red photogate of the type used in
general education laboratories was mounted so the flag would trip the photogate during pendulum oscillation. Two different timing measurements were then
performed, using a
In the pendulum mode of the timer, the period was directly measured. For this case, the photogate was positioned, relative to the U shaped flag (for which one vertical arm is slightly longer than the other), so as to be interrupted only once by the pendulum per pass. In the time-interval mode, the flag was positioned so that both arms interrupted the photogate beam. The reciprocal of this time of interruption proved to be a reasonable measure of the instantaneous speed of the pendulum at the position of the photogate, which was that of maximum kinetic energy. The time intervals were recorded manually for traversals separated by one period, through 5 cycles of oscillation. These numbers were then typed into Excel and their reciprocals graphed. A trendline (using option to print the slope) was applied to the near linearly declining graph. The decrement of this line (fractional decrease per cycle) proved to be a good approximation to the logarithmic decrement of the motion, which could have been estimated with exceptional precision by means of the other techniques mentioned in this chapter.
In the first set of experiments, the sphere on the lower shaft was maintained at a constant distance from the knife-edge, while the mass on the upper shaft was positioned at increasingly greater distances from the knife edge to lengthen the period. Over the full range of periods considered, the distance between the two masses changed by a small amount around 67 cm. The results of this first study are shown in the left graph of Fig. 14, where the log-decrement has been plotted versus the square of the period. The Q of the pendulum (p /D) may be calculated for any value of the period using the indicated slope of 0.0004. For example, the Q at a period of 10 s was 76; this being near the shortest period considered. Near the other extreme of T =35 s, Q = 6. At the shortest possible periods, damping due to air drag would begin to become important.
Figure 14. Results of experiments to test the
dependencies of Q on (i) frequency and (ii) length of
pendulum
The
reasonable fit of the linear regression vs period
squared is consistent with the prediction by equation (31) that Q should be quadratic in the
frequency.
The equation (31) also indicates that the
log-decrement should be proportional to the reciprocal of the distance L
between the masses. To test this
prediction, the period of the pendulum was measured as a function of mass
separation, also using the smart timer.
In generating the data for the right graph of Fig. 14, the period was
maintained constant at 20 s. For every
datum, the top sphere was always only slightly closer to the knife edge than
the lower sphere. At 0.049, the
intercept of the trendline differs enough from zero,
relative to the size of the error bars, to imply a systematic error. Possible sources of the error include: (i) the masses are
of finite size, rather than being points as assumed by the model, and (ii) a
non-negligible mass from parts other than the spheres. Nevertheless, the data show a clear size
dependence of the Q.
The experiments just described do not permit one to readily isolate the source of the damping, which for the cases of Fig. 14 had the knife edge resting on silicon wafers (integrated circuit stock material). It is not known to what extent the dissipation was dominated by strain in the knife edge/silicon interface or by flexure of the aluminum arrow. Although the model which generated equation (31) assumed only the latter, there is nevertheless theoretical and experimental basis for model acceptance, regardless of the details of the damping.
12.3.4 Highly dissipative, yet hard materials
The same pendulum was used to demonstrate some counterintuitive features of internal friction damping, by replacing the Si wafers with various materials. When very soft material, such as lead, was the support for the knife-edges, there was a significant increase of the damping, as expected. It was also found, however, that cast iron increased the log-decrement (10 s period) by more than 40% . The same was also true of ceramic PZT wafers, of the type used to ignite gas grills by striking the wafer impulsively. Both the cast iron and the lead-zirconate-titanate samples are very hard, so the internal friction must derive from large defect densities in which atomic disorder is a sensitive function of stress. Some other hard interfaces, such as steel on glass, or steel on sapphire, did not show a difference from steel on Si; which suggests that the dominant source of damping for the pendulum in all these cases was flexure of the aluminum shaft.
The observation involving cast iron is consistent with its known excellent damping properties at higher frequencies--important, for example, to engine blocks. Some magnesium alloys have also been developed that have excellent damping characteristics without seriously sacrificing strength.
13. Internal Friction
13.1 Measurement and Specification of Internal Friction
Mechanical spectroscopy is a popular means for measuring internal friction of materials. [Fantozzi 1982]. Typically, a torsion pendulum is used to harmonically stress a sample and the lag of the response (strain), relative to the stress, provides the loss tangent and thus the internal friction. In such experiments it is widespread practice to report internal friction as Q-1. There can be confusion because of this practice, depending on the nature of the measurement technique; i.e., whether one actually measures Q as opposed to measuring something proportional to the stress-strain lag angle. If Q is obtained from an oscillatory free decay, using the logarithmic decrement defined as follows, then there is no problem.
(32)
Here x n and x n+1 are adjacent turning point amplitudes separated by one period of the motion, T. In practice it is very difficult to adjust a mechanical system to oscillate over a wide frequency range. The widest range known to the author, for a mass-spring system, involved the work of Gunar Streckeisen , in which a vertical seismometer using the LaCoste spring was adjusted to have periods in the range between 7 s and 140 s. Because of the difficulties in attaining a wide range of eigenmodes, internal friction is typically determined with a specimen that does not oscillate. We now consider that case.
13.2 Non-oscillatory Sample
In the typical torsional pendulum used to measure internal friction, the sample is of very small mass. Such a pendulum was built, for example, around the original version of the fully differential capacitive sensor, to study magnetoelastic wires [Atalay 1992]. As with many delicate instruments, the Atalay & Squire instrument was of the type labeled "inverted". A silk fiber at the top of the specimen was used to provide minimal tension in the sample. They used one LRDCT [Peters 1989] in the drive mode to provide a known stress to the delicate magnetoelastic sample and a second LRDCT to measure the strain magnitude and the angle by which it lags behind the stress, because of anelasticity. As such they were measuring the lag angle and not Q, as will now be shown.
Without an inertial term, the sample response x to a periodic external force F is governed by
(33)
so that the transfer function is given by
(34)
from which it is seen that the measurement does not yield Q-1 but rather the lag angle z / k , where k is constant. Perhaps the measured angle, which is an indicator of the internal friction, has been called Q-1 because k = mw02 for an oscillator of frequency w0 , and Q = mw02/z for the freely-decaying oscillator. Bear in mind, however, that this expression for k does not apply to the non-oscillatory measurement just described. There is a frequency square difference between such a measurement and what would be measured if an adjustable oscillator were being considered.
An example of the importance of this issue is found in the article [Lakes, 1996] in which one reads from the abstract
the
following statement: "The damping, tan d, followed a n- n dependence, with n » 0.2, over many decades of
frequency n. This dependence corresponds to a stretched
exponential relaxation function, and is attributed to a dislocation-point
defect mechanism. It is not consistent with a self-organized criticality
dislocation model which predicts tan d µ An-2. Dislocation damping in metals is relevant to development of high
damping metals, the behaviour of solders and of
support wires in Cavendish balances."
The present
arguments suggest that the experiment by Lakes and Quackenbush
is not in strong disagreement with the SOC model; that the magnitude of the
exponent difference between theory and experiment is really 0.2 and not 1.8 as
they have indicated.
13.3 Isochronism of Internal Friction Damping
It is well known that in the viscous damping free-decay case, the frequency of oscillation is lowered by damping according to
(35)
and the resonance
frequency of the driven oscillator is lowered even further [
(36)
where sgn(dx/dt) is the algebraic sign of the velocity--it causes the equation of motion to be nonlinear even if the square root term were not present. For small damping, the square root term can be shown to be equal to the time dependent amplitude of the motion multiplied by the angular frequency.
Other damping types are possible and are indicated in [Peters 2002,.. universal...] where also is provided evidence for harmonic distortion in the waveform because of the nonlinearity. It is shown in [Peters 1997] that the oscillation is isochronous.
For large values of Q, the lag angle (radian measure) is given by d = 1/Q. Researchers usually measure d and specify the magnitude of the internal friction as Q-1. As noted previously, Q is proportional to frequency for the viscous damped oscillator. Thus, for viscous damping, the internal friction is inversely proportional to the frequency.
For hysteretic damping we obtain the result
(37)
where the variables are defined in equation (19). For small damping in which tan d = d = Q-1, we find that the internal friction for hysteretic damping is inversely proportional to the square of the frequency, since h is constant and k = mw2.
14. Mathematical tricks -- linear damping
approximations
14.1 Viscous damping
In the Hooke’s law expression, F = -kx, it is common practice to approximate hysteresis of oscillatory motion by letting k become a complex coefficient. This is also standard practice in a variety of fields, such as the description of lossy electromagnetic media. No doubt the practice has been further popularized by the standard approach of solving electrical engineering a.c. circuit problems by means of phasors, the technique developed by Steinmetz.
We recognize in the expression x(t) = x 0 e jwt = x 0 cos wt + j x
0 sin wt that harmonic variation is contained in the
real part (or alternatively the imaginary part) of the complex exponential
form. Using
(38)
where the
approximations, 
have been
employed.
But
since 
, equation (38) can be
rewritten as
(39)
We thus see that the damping constant 
permits us to express
the logarithmic decrement D in terms of the angle d with
which x lags F; i.e., 
. (Note that we are
making no distinction here between the period with and without damping; since
the difference is small and hard to measure.) If b were independent of frequency then d would
be inversely proportional to the frequency, which is rarely realized in
practice.
14.2 Hysteretic damping
Equation (39) does not properly represent some of the most important engineering systems. For those labeled ‘hysteretic’, we must use a different form for the complex spring constant. We assume that F = - (k complex )x = (k + jh)x where h is a real constant. Since dx/dt = jwx, this yields the equation of motion
(40)
Since h is assumed to be a true constant (independent of frequency) the lag angle between displacement and force is given by
(41)
which is seen to be inversely proportional to the square of the frequency. (Note that d here is the same as a in Fig. 6.) It should be noted that the complex form for the spring constant is not simply obtained using the common theory of viscoelasticity. Such theory requires a multitude of relaxation times (stretched exponentials). [ Speake 1999].
15. Internal Friction Physics
15.1 Basic Concepts
Virtually all damping derives from varying degrees of complexity, because of the myriad interactions that are present, either internal of nonconservative type of external so as to involve environment. This is true even for systems that come closest to being governed by the textbook equations . For example, the author has attempted to produce ideal harmonic oscillators using viscous liquids for damping. Even they are complicated and do not strictly obey Stokes’ law of drag force proportional to the velocity. The nonlinear Navier Stokes equation may be capable of describing them, but not in a simple form except to a first approximation. This approximation is not really very good—relative to the precision that is possible with modern sensors.
Perhaps the closest to being an
ideal viscous damped oscillator is that in which the damping force derives from
eddy currents through Faraday’s law. A
magnet is attached to the oscillator, and as it moves in proximity to a condutor; the time rate of change of magnetic flux gives
rise to a retarding force that is propotional to
velocity. Because there really is a
force involved, and because of Lenz’ law, the damping term makes sense
physically. This case might be
completely ideal except for one factor—the magnet is part of a mechanical
system that must possess structural integrity if it is to oscillate. Because of loads present in the structure
(reactionary normal forces to the various weights), there will always be some
creep. The creep is ultimately
unavoidable, since there is apparently no stress threshold below which plastic
deformation ceases to exist. It is
important to realize that forces associated with inertial mass (
15.2 Dislocations and Defects
It is surprising the extent to which mechanical defects, such as dislocations, have been ignored by large segments of the scientific community. The surprise is even greater, when one considers the importance of defects in another field—that of electronics. Our present information age (world of computing) came into existence only after widespread recognition of the importance of the defects called impurities. The n-type and p-type semiconductor materials necessary to our modern age result from the substitution of silicon atoms with others of pentavalent and trivalent type in surprisingly small concentrations.
The strength of solids is very much less than as predicted by theories of an ideal (perfect) crystal. The primary culprit is dislocations. Their influence on materials used in engineering has prompted the statement “when mother nature fills the vacuum she abhors, she rarely does so with perfection”. Unfortunately, few students exposed to fundamental science get training in defect physics. Moreover, it is difficult to provide a self-consistent fundamental description of their properties; so that few scientists have more than a superficial knowledge of their importance.
Viscoelasticity is a misleading term. To combine the words ‘viscous’ and ‘elastic’ suggests that the state variables vary smoothly in time; i.e., as a fluid in the viscous part. Unfortunately, this is not true of hysteresis associated with either ‘domains’ or with ‘grains’. In the case of magnetic domains, it is quite easy to demonstrate non-smooth (jerky) behavior that is called Barkhausen noise. Although the phenomenon was demonstrated by Barkhausen in 1919, only recent studies have begun to better understand some of its complexities. [Urbach 1995].
A similar phenomenon that must relate in some manner to the Barkhausen effect is the PLC effect. Under applied stress, alloys frequently display discontinuous strain increase (jumps). The author has even demonstrated strain recovery of similar type, catalyzed by ‘tapping’. The polycrystalline metals that demonstrate these effects are obviously influenced by ‘granularity’. They differ from the ‘granular materials’ that have become a ‘hot’ topic of recent research. Even pure polycrystalline metals exhibit these features. The German word to describe the deformation of tin under large stresses is ‘zinngeschrei’ = ‘tin cries’. Anyone who has ever bent large diameter tungsten wire has experienced this phenomenon, since the non-smooth strain can be both felt and heard.
There is still another type material, thought to have great engineering potential in the future, that shows ‘granular’ behavior—that of shape memory alloys (SMA). If an SMA specimen is cycled in temperature around the martensitic phase, it generates acoustic emissions [Amengual 1987]. For a figure taken from their work and other good pages about hysteresis, refer to the webpages of Prof. Sethna at http://www.lassp.cornell.edu/sethna/hysteresis/ReturnPointMemory.html] These emissions are probably related to the Portevin LeChatelier effect and are characterized by surprising reproducibilities in spite of their complex behavior.
Thus there is abundant experimental evidence against the overly-simplistic view that hysteretic damping can be meaningfully described by simple, linear differential equations. The nonlinear terms necessary for a good mathematical treatment go beyond ‘chaos’ to the world of ‘complexity’. Chaos of deterministic type, though bewildering to many, is in many cases tractable (using equations that can be integrated numerically). Damping problems are much more complex than deterministic chaos. The challenges to our understanding derive in part from the long time that it has taken before there were any serious investigations of the mesoscale, the place where defect structures abide. If as with Zener we use the word ‘anelasticity’ to describe systems that are ‘other than’ elastic; then the term mesoanelastic complexity is an appropriate label for this poorly understood physics that is important and yet mostly unknown to many fields of both science and engineering.
16. Zener Model
16.1
Assumptions
The standard linear model of viscoelasticity provides a sound basis for some damping phenomena, yet it fails badly as an approximation for hysteretic damping. Its prominence in both the worlds of physics and engineering warrants the following detailed discussion, so that the failure case may be properly documented.
Following the example of Zener, the following linear differential equation relates the stress s, the strain e , and their first time derivatives.
(42)
The t’s are
relaxation times (subscript e meaning at
constant strain and subscript s at constant
stress), and E1 is the relaxed elastic modulus (ratio of stress to
strain in a very slow process. Nominally, ts > te
, consistent with strain lagging stress. For periodic variations
(43)
which, when
substituted into equation (42) yields
(44)
The complex modulus of elasticity is defined by
(45)
and is seen to
relate stress and strain according to
(46)
From equation (45), the real and imaginary parts of
the modulus are found to be
(47)
(48)
The independent variable, or ‘frequency’ for all
cases is the convenient dimensionless parameter, 
.
It is convenient to use polar form, so that
(49)
where 
is obtained by
computing the square root of the sum of the squares of the real and imaginary
parts. In this form it is apparent that d is a lag angle which determinines
the damping loss for the system.
Moreover, from equations equations (47) and
(48) it is seen to obey
(50)
16.2 Frequency Dependence of Modulus and Loss
The essential
features of the Zener model are illustrated in Fig.
15, where the ‘unrelaxed’ high frequency modulus
obeys the relation 
Figure 15. Zener Model of anelasticity. Bottom curves are ‘frequency’ variation of modulus and loss respectively.
In viscous damping models, the damping is quantified
by the product bT, which is
equal to the logarithmic decrement. The
logarithmic decrement is directly proportional to the period when the damping
‘constant’ b is truly
constant. The graph in Fig. 16 compares
the logarithmic decrement computed by the standard model against a case where b = constant. Also shown in the
figure is a set of hysteresis curves for wt = 10, 1, and 0.1 respectively. Notice that the damping is large only for wt near 1, in
accord with the bottom plot of Fig. 15.
For that case, points ‘a’ through ‘f’ and back to ‘a’
are shown, labels to illustrate work done by the stress in traversing the hysteresis loop. The
algebraic sign of the work changes around the loop and the net work done in one
cycle is just the area enclosed by the loop. 
Figure 16. Characteristics
of the Zener Model.
For damping based on the Zener
(standard linear) model to agree with the simple viscous approximation, it is
necessary that wt >>1;
i.e., the period of the oscillator must be significantly shorter than the
smaller of the relaxation times, as illustrated in the bottom graph of Fig. 16.
16.3 Successes--models of viscoelasticity
Viscoelasticity, as an approximation for damping, is evidently
quite adequate for some materials. The
assumption of fluid character as a basis for hysteresis
is expected to be closest to correct when applied to those cases in which
variations in strain are almost continuous.
The materials of rheological type for which
this appears to be most true are solids built from long chain polymers; i.e.,
various plastics. Such materials can
yield surprising results, however. Shown
in Fig. 17 are results from a study that used a nylon monofilament sample (8 lb
fishing line). The pair of torsional free decay records correspond to two different
temperatures--290 K (room temperature) and 390 K (above the glass transition
temperature of the nylon). Although a
significant increase in the period was observed as the temperature was
increased above the glass transition temperature (changing from 18.2 s to 27.8
s), the logarithmic decrement was found to be almost unchanged. This was not in keeping with the expectation
that softening of the material at the higher temperature would result in
significantly greater damping. The
effect is just the opposite of what was mentioned concerning cast iron, which
though very hard, does not have small damping.
Here, a softening does not result in significantly increased damping.
Figure 17. Torsional
free decay records of monofilament nylon at temperatures first below and then
above the glass transition temperature.
Although the modulus decreased dramatically at the higher temperature,
the damping did not.
Although there was some creep observed for both the decays of Fig. 17, the creep was more pronounced in the higher temperature case. This is illustrated by the lower curve of the bottom graph, which is a computer fit in which the secular term necessary for best fit was removed to illustrate the creep. In both decay cases, the log decrement was calculated by importing the A/D data (Dataq DI-154RS) to Excel and then using trial and error adjustment of parameters to achieve the best fit.
Although the damping of glasses is normally treated using the theory of viscoelasticity, Granato has recently modelled these materials via defects. In his paper [Granato 2002] he states the following: "As dislocations carry the deformation in crystals, interstitials are the basic microscopic elements carrying the deformation in glasses near and above the glass temperature".
16.4 Failure of viscoelasticity
Unfortunately for the elegant theory of the Zener model that has been presented, there are many mechanical
systems for which the Q is not proportional to frequency,
but rather proportional to the square of the frequency. The logarithmic decrement (D = p /
Q) has been measured for a host of long-period mechanical oscillators,
configured as some form of a pendulum.
In all cases these systems were described approximately by D = bT a T2 rather than by bT a T. Similar behavior has been noted in mechanical
oscillators other than the pendulum--for example, in the geophysics research of
Gunar Streckeisen and
Erhard Wielandt; who are well known for the
development of the widely employed STS-1 seismometer. During his pursuit of the PhD [Streckeisen 1974], Streckeisen
measured the numerical damping (fraction of critical damping) of a vertical Sprengnether long-period seismometer 5100-V. After removing the magnet of the velocity
transducer (to eliminate eddy currents and reduce viscous air damping, he found
that the numerical damping was proportional to the square of the period between
periods of 7 and 140 s. He took about 30
measurements over this interval of periods; and showed that the damping
increased from about 0.0008 to about 0.3—a factor of roughly 400, not 20 as one
would expect for viscous damping. To
quote Wielandt (private communication), “the data are
very clear”.
17. Toward a Universal Model of Damping
17.1 Damping capacity quadratic in frequency
The quadratic dependence on frequency of Q (log decrement proportional to period squared) is equivalent to friction force being frequency independent. In support for the claim of universality, it was noted in the introduction to this chapter that three very different systems showed this characteristic: (i) the vertical seismometer just discussed, (ii) various pendula, and (iii) the rotating rod direct measurement of internal friction first done by Kimball and Lovell; who measured the transverse deflection of the end of a rod when it was rotated about a horizontal axis.
17.2 Pendula and universal damping
An example of one of the author's experiments
that illustrate universal (hysteretic) damping is provided in Fig. 14. Other works that illustrate hysteretic
damping include those by Peter Saulson of
The pioneering work of Braginsky
(important to LIGO) has already been mentioned in the context of small force
measurements and noise. He and his
An oft-cited paper speaking to the issues of hysteretic damping is an article by Quinn et al [Quinn 1992]. Concerned with material problems in the construction of long period pendulums. (The type of pendulum on which they based their studies was first described in the scientific literature two years earlier [Peters 1990].) In a follow-on paper, Speake et al state the following: "The analogues of anelasticity and its resultant 1/f noise are seen in a wide range of other processes (for example, dielectric and magnetic ones) described in terms of frequency-dependent susceptibilities". [Speak 1999].
The jerkiness (discontinuous change) that is the hallmark of the Barkhausen effect may have been first seen mechanically in the experiments that generated the metastable states paper. From a consideration of the chapter by James Brophy [Brophy 1965]], it was postulated in this 1990 paper that the jerky behavior of a mesodynamic pendulum is a type of mechanical Barkhausen effect.
17.3 Modified Coulomb Model--Background
The results which follow grew naturally out of the application of fully differential capacitive sensors to the study of mechanical oscillators. Efforts to model internal friction inluence on long-period pendula uncovered something surprising to most--that the foundation for physics laid by Charles Augustin Coulomb may be much broader than had been realized. Most individuals in the physics community do not associate Coulomb's name with contributions other than to the laws of electrostatics. Engineers, however, have long used his name in the context of sliding friction; since, in fact, Coulomb gave us the empirical description which involes static and kinetic coefficients. Because of his interest in the civil engineering of soils [Heyman 1997]. Coulomb also provided something else--a basis for understanding granular flows and even some types of fracture. Concerning the latter, the Mohr criterion, applied to the Coulomb failure envelope, defines a 'coefficient of internal friction' which is used to predict brittle failure [Gere 1996].
Coulomb friction is empirically simple, at least as a first approximation; since it depends only on the normal force between surfaces and the algebraic sign of the velocity, when there is relative motion. Like so many problems of many-body type, a complete theory of sliding friction is far from being realized. Simplistic textbook efforts to explain energy loss, by picturing 'hills and valleys' of the surface of two solids in contact, are useless. An example of such naivete can be realized by trying to understand the phenomenon of optical contacting. Two orthogonally oriented fused silica cylindrical fibers, allowed to touch, can experience atomic bonding forces that are surprisingly strong, being much greater than the weak attraction of van der Waals type. Cleanliness of the surfaces is paramount for success in such a demonstration, which speaks to another issue--a connection between internal friction and surface physics.
The conversion of mechanical energy to thermal
energy must involve nonlinear (avalanche or cascade) processes. A heuristic description of defect structural
interactions that generate heat and eventual failure is the phonon triangle of
Tom Erber (
Figure 18. Heuristic description of how materials
fail--processes connected with damping
There must first be a downward path to the micro-level, through cascading. These cascades can cause Barkhausen noise in the case of ferromagnetic samples, and acoustic emission in nonmagnetic metallic alloys (Portevin LeChatelier effect). Failure requires the upward path of defect organization, the mechanisms of which are not yet understood. One of the first theories with possible implications to the organization leg is that of self organized criticality. In the magnetic case, Erber has used a fluxgate magnetometer to improve failure predictions; since magnetic hysteresis is proving to be a sensitive indicator of mesoscale structure changes during cycling toward failure. Inferred from these studies is some yet-to-be discovered connectivity between noise, damping, and failure.
Surface friction is expected to somehow be connected with internal friction--the biggest difference being that the surface has many more defect states with which to redistribute energy. The larger density of states of the surface (reduced order) is probably an important factor in the difference between surface friction and the modified internal friction model which follows.
17.4 Modified Coulomb Damping Model --Equations
In
the following damping model for internal friction, Coulomb’s law of sliding
friction is modified by assuming that the coefficient of friction is not
constant, but rather involves the energy of oscillation E in a power law; i.e.,
(51)
where
c = constant that is different for each l. For Coulomb
(sliding) friction l =
0. For amplitude-independent damping of hysteretic
type, l
= ½. For amplitude dependent
(such as large Reynolds number fluid) damping l = 1. In all cases, if c <<1
(small damping), the damping capacity is quadratic in the frequency, so that
the internal friction Q-1 ~ w-2. Equation (51) is easily implemented, in spite
of its nonlinearity; which we will see later to be cause for harmonics in the
decay.
It is convenient to rewrite equation (51) in canonical form, so as to involve the Q of the oscillator. For the case of hysteretic damping (l=1/2), the equation becomes
(52)
Similarly, for amplitude dependent damping (l = 1)
(53)
where y0 is the initial value of the amplitude of x (largest maximum of x); and Qf is found not to be constant, as in the case of hysteretic damping. Rather, in this case the Q increases as the amplitude decreases. On the other hand, the Q of an oscillator influenced only by Coulomb (l = 0, sliding) friction decreases with the amplitude; and the equation of motion in canonical form is given by
(54)
In equations (53) and (54) the subscript 0 is used to identify the initial value of the time varying Q . Equation (54) is equivalent to equation (12) with Qc0 /y0 = p /(4Dx).
As will be illustrated with some examples, it is possible for an oscillator to be influenced simultaneously by all three types of friction. One may treat such a system with the following equation of motion
(55)
At any instant during the decay, the total (time dependent Q) is given by
(56)
in which it is seen that the smallest Q in the set (largest damping term) is dominant, in a manner reminiscent of capacitors connected in series.
It is instructive to look at the analytical solution for the time dependence of the amplitude (turning points, y(t) = |xmax|), when all the Q's >> 1. Such a solution is obtained from energy considerations, by noting first that the time rate of change of the energy is zero in the absence of friction; i.e.,
(57)
With friction, dE/dt is determined by the rate of doing work against the friction force; i.e., dE/dt is proportional to w y f, where f is the friction force. In the case of Coulomb friction, f is constant (determined by y0, so dE/dt is proportional to E1/2. For hysteretic damping, f is proportional to y, so dE/dt is proportional to E. For fluid damping, dE/dt is proportional to y2 , so dE/dt is proportional to E3/2. Thus, the general case is described by
(58)
Because the energy is proportional to y2 we can write down the equation for the time varying amplitude as
(59)
where a, b and c are constants. The solution to this first order equation can be found in integral tables, and the result depends on the size of c relative to the product ab. For present purposes, we will restrict ourselves to the case where Coulomb damping is not dominant, in which the solution involves an exponential. (For large c, one may develop the corresponding general case in terms of the tangent or its inverse.) The present result is as follows, using r = (b2-4ac)1/2, where 4ac < b2 :
(60)
In the case where c = 0, equation (60) can be simplified to the following form, which is useful for curve fitting:
(61)
For the case where a = 0, the better form for curve fitting is:
(62)
17.5
Model output
Shown in Fig. 19 is a
case in which the decay is influenced by all three types of friction. Notice how the Q rises initially, peaks at a
value less than what would be true for hysteretic damping alone (constant Q
case), and then later declines. The
initial rise is due to the amplitude dependent damping term (size determined by
coefficient a), and the later decline is due to the Coulomb damping term
(determined by coefficient b).
The code in Table I which was used to generate Fig.
19 has been reproduced here for two
reasons: (i)
to show the ease with which the modified Coulomb model may be numerically
applied in general to a damping problem, and (ii) to illustrate an integration
algorithm that has proven to be intuitive, simple, and powerful--the
Cromer-Euler technique which Alan Cromer first described as the 'Last point
approximation (LPA)' [Cromer 1981] in contrast to the unstable 'first point
approximation' given to us by Euler.
Over the last twenty years the author has employed the LPA in a host of
applications that span from the generation of satellite ephemerides
in the
Figure 19. Model generated results based on equations (55) and (60).
TABLE
CLS
REM: setup display
SCREEN 12: VIEW (0, 0)-(600, 470): WINDOW
(-.2, -5)-(1, 5)
REM: assign constants and initialize variables.
pi =
3.1416: dt = .002: t = 0
x0 =
4: x = x0: y0 = x0: xd
= 0
period =
.5: omega = 2 * pi / period: b = .1: a = .1: c = .01
REM: print damping coefficients
PRINT " DAMPING COEFFICIENTS: a = "; a; ", b = "; b; ",
c = "; c
r = SQR(b ^ 2 -
4 * a * c): alpha = 2 * a * x0 + b - r
beta = 2
* a * x0 + b + r
REM: Use a,b &
c--set Q's to dampen (quadratic, linear, & constant resp.)
qf =
omega / 2 / a / y0: qh = omega / 2 / b: qc = y0 *
omega / 2 / c
REM: start integration loop
LOOP0:
t = t + dt
REM: analytically compute amplitude (y = magnitude of x) at each time
point
p = alpha * EXP(-r
* t) / beta
y = (b * (p - 1) + r * (p + 1)) / 2 / a /
(1 - p)
REM: integr. the eq. of motion to get x(t), using 3 fric.
force/mass terms
REM: The coeff.'s ff, fh
& fc correspond to: quadratic in speed (fluid),
REM: linear in speed (hysteretic), and independ. of speed (Coulomb) resp.
ff =
(pi / 4) * (1 / y0) * (1 / qf) * (omega ^ 2 * x ^ 2 +
xdot ^ 2)
fh =
(pi / 4) * (omega / qh) * SQR(omega ^ 2 * x ^ 2 + xdot ^ 2)
fc =
(pi / 4) * omega ^ 2 * y0 / qc
REM: check algebraic sign--USE SIGN BUT NOT MAGNITUDE OF VELOCITY
IF xdot > 0
THEN GOTO SKIP
ff =
-ff: fh = -fh: fc = -fc
SKIP: xdoubledot
= -ff - fh - fc - omega ^ 2
* x
xdot = xdot + xdoubledot * dt: x = x + xdot * dt
REM: calculate the energy and then the amplitude to evaluate Q
REM: could instead use analytical result q = (pi/4)*omega^2*x/abs(ff+fh+fc)
energy =
.5 * xdot ^ 2 + .5 * omega ^ 2 * x ^ 2
amplitude =
SQR(2 * energy) / omega
REM: calculate loss per period due to friction
loss =
ABS(ff + fh + fc) * 4 *
amplitude
q = 2 * pi * energy / loss
IF t < 1.2 * dt
THEN PRINT " initial Q = "; 10 * INT(q) /
10;
IF t < 20 THEN GOTO SKIP2
PRINT ", initial Amplitude = ";
x0; ", Period = "; period;
PRINT ", final Q = "; 10 * INT(q) / 10
REM: DO GRAPH
SKIP2: PSET (.04 * t, .5 * q / omega):
PSET (.04 * t, .5 * qh / omega), 4
PSET (.04 * t, 4 * x / y0): PSET (.04 *
t, 0): PSET (.04 * t, 4 * y / y0)
IF t > 20 OR y < 0 THEN GOTO pause
GOTO LOOP0
pause: GOTO pause
RETURN: END: STOP
17.6 Experimental Examples
The code of TABLE I is useful in determining the
nature of a given experimental case. Too
frequently in the past, it has been naively assumed that the entire decay
record was exponential. Particularly
when longer records are collected, it is found that most damping is
nonlinear. In the two experimental
examples that follow to backup this claim, one is a near perfect (nonlinear)
particular case of amplitude dependent (fluid) damping, and the other is a
mixture of amplitude-dependent and hysteretic damping, but devoid of any Coulombic influence.
Coulomb friction frequently tends to be either 'all' or 'nothing',
depending on whether there is an unwanted mechanical contact that involves
slippage. A notable exception is found
in the case where a pendulum is influenced by eddy current damping in a narrow
region of its total motion [Singh 2002].
The pendulum
that was used to generate the data displayed in Fig. 14 was also used as
follows. A large flat piece of plastic
was attached to the bottom of the pendulum, so that its movement (normal vector
to the surface in the direction of motion) would disturb a great amount of air
in turbulent manner. As expected, there
was a dominant initial (large level) amplitude-dependent damping, as shown in
Fig. 20.
Figure 20. Decay
of an air damped pendulum as a function of cycle number n.
The speed (maximum) was measured with a photogate as previously discussed. The expressions shown in the figure are consistent with equations (59) and (61). The data, which were collected by hand and typed into Excel, produced the 'jagged' curve; and the computerized fit according to equation (61) is the smooth curve of the pair. It is noteworthy that the quadratic drag of the air (determined by a = 0.036) is 40% greater than the viscous drag at the start of the decay. By cycle 37 the quadratic part has become much less significant than the constant Q viscous part -- having become roughly 60% smaller.
The fluid damping 'soupcan' pendulum data of Fig. 21 was generated with a can of Bush's blackeye peas. The container with enclosed unbroken contents, being a right circular cylinder of length 11. cm x diameter 7.4 cm, was suspended horizontally by a pair of knife edges under opposing end-lips [Peters 2002, Soupcan pendulum]. The motion of the can was measured with an SDC sensor connected to a Dataq A/D converter. Whereas experiments of similar type, with homogeneous fluid contents, have produced viscous decay records; the present case involved only friction of so-called 'fluid' type; i.e., quadratic in the 'velocity'. To generate the figure, the A/D record was exported to the Microsoft software package, Excel. Fits to the data were then obtained by adjusting, through trial and error, the a, b, and c coefficients of a 'fit' to the amplitude. For this case the fit was easily accomplished because both b and c proved to be essentially zero.
The second case, involving an evacuated pendulum, was not a single pure type of damping, but can be seen in Fig. 22 to have both hysteretic and amplitude dependent contributions. Although fluid damping is amplitude dependent in the same manner, with the damping term being proportional to the square of the amplitude; the word 'fluid' is not used to describe this case since the system involved exclusively solid materials.
Not all decay records of this pendulum in vacuum yielded a mix of friction types as displayed in the figure. The effect was observed to be transient, and it is speculated that outgassing of components may have been a factor.
Figure 21. Example of fluid damping of a 'soup-can' pendulum. The granular contents (blackeye peas and water) results in a friction force that is quadratic in the velocity.
Figure 22. Example of a mix of two damping types, hysteretic and amplitude dependent.
Numerical
Integration
Instead of integrating the second-order equation of motion twice—first the acceleration, followed by the resulting velocity; more accurate results are obtained by integrating the equivalent pair of first order equations.
For example, the equation of the simple harmonic oscillator with viscous damping is expressible as
(63)
where the position variable has been represented by the generalized coordinate q (x elsewhere), and for the momentum p = m dq/dt, and here the mass, m, has been set to unity. Likewise, the spring constant has been set to unity. It is generally useful to distill a given problem to its most basic form when attempting to understand the physics. Constants that provide no useful information for trend analysis purposes are conveniently ‘normalized’. Such is common practice, for example, in modelling chaotic systems.
The second-order set can always be reduced mathematically to a first-order pair; however, the pair results naturally from the use of the Hamiltonian as opposed to the Newtonian formulation of mechanics.
17.7 Damping and Harmonic Content
Equations (52) through (54) are
the nonlinear, modified Coulomb damping model forms that correspond
respectively to (i) hysteretic, (ii)
amplitude-dependent, and (iii) Coulomb damping.
The damping term for each of the three cases can be expressed as
follows:
(64)
where f is the friction force, and [v] is
the square wave whose fundamental in a Fourier series expansion is equal to the
velocity of the oscillator times 4/p.;
i.e., for a square wave + h, the
amplitude of the fundamental is + (4 /p
)h. We see that all the damping types
that have been considered in this chapter, when expressed in canonical form,
correspond to a fundamental friction force
f
= m w v / Q.
The simplicity of this result is probably why viscous damping has been
viewed by so many physicists as 'inviolate'.
One must be careful, however, because as noted in the previous section;
only for the case of hysteretic damping is Q constant. For amplitude-dependent damping Q f
= Q f 0
(y0 / y ) and for Coulomb damping Qc = Qc 0 (
y / y0). The time
dependent Q of non-exponential cases will have significant influence on mode
development in many body systems, because of elastic nonlinearity (necessary
for mode coupling).
There
is another important subtlety of equation (64).
When only the fundamental of [v] is retained, equivalent to viscous
damping, Q is
proportional to frequency. When all odd
harmonics are included (full square wave), Q becomes proportional to frequency
squared. This means that
harmonics in the friction force are responsible for the primary difference
between hysteretic damping and viscous damping. One of the things being presently considered
is how, in an algorithmic sense, to modify equations (52) through (54) to
provide for 'dispersion'; i.e., means
for providing Q dependence other than frequency squared . We posit the following--hysteretic
(exponential) damping is the idealized universal form of damping, due to
secondary creep. When there is an
activation process of Zener (Debye)
type, such as dislocation relaxation; then additional terms must be added to
the hysteretic 'background'. It may be
this can be accommodated by a suitable removal of harmonics from the square
wave of the hysteretic case, and it may happen that Q is constant for systems
that vary continuously. It is
conjectured that the PLC effect, responsible for discontinuous changes, plays a
role in those cases where Q is not constant.
18. Nonlinearity
18.1 General considerations
Electrical
nonlinearity is the type with which most engineers are familiar. It is the very basis for common non-digital
forms of communication, such as that of frequency modulation type. A popular form of radio amateur communication
is one in which the carrier and one of the two normal sidebands of a signal are
suppressed before going to the antenna.
At the receiver, the carrier is 'regenerated' before going to the demodulator. The demodulator required for ultimate
transduction by speaker is also a nonlinear device.
Nonlinearity
of mechanical type is encountered throughout nature. The human ear, for example, is not linear,
but rather characterized by both quadratic and cubic nonlinearities. If an intense, pure low frequency (inaudible)
sound of frequency f is present with a higher frequency audible one of
frequency F; then one typically hears (in addition to F) tones at F +/- f due
to the quadratic nonlinearity and F +/- 2f due to the cubic nonlinearity.
Very
high frequency acoustics (ultrasound) is employed for studies of
elasticity. The quasi-linear features of
ultrasonic propagation have been the basis for measuring 2nd order elastic
constants (determined by velocity of propagation) and internal friction (by
attenuation of the beam; i.e., damping).
A
commonly employed ultrasonic technique that has been used to study both linear
and nonlinear phenomena is the pulse-echo method. By using a thin specimen and extending the
pulse width, the overlapped signal can add constructively or destructively; and
in the former case, resonance is approached as the width gets very large
[Peters 1973]. The pulse-echo method was the basis for this author's PhD
dissertation ["Temperature dependence of the nonlinearity parameters of
copper single crystals", The Univ. of Tenn., 1968]. The distinguished career of his professor, M.
A. Breazeale, has focused on ultrasonic harmonic
generation as a means to determine the shape of the interatomic
potential of solids [Breazeale 1991]. A longitudinal wave distorts because of the anharmonic potential (acoustic equivalence of optical
frequency doubling with lasers in a KdP
crystal). In like manner, phonon-phonon
interactions are possible only because of non-zero elastic constants of order
higher than 2nd (2nd order constants determining the harmonic potential). Because phonon-phonon interactions are part
of damping, there must be consequences, at least for some cases, from nonlinear
damping terms.
The
unifying theme for this chapter is that damping is fundamentally nonlinear, in
spite of the fact that linear approximations have prevailed in modelling; and for many purposes these linear models appear
to be acceptable [
Although
their statement may be true for steady state, it is not expected to be true for
the transient processes that lead to steady conditions of oscillation. As demonstrated elsewhere in this document,
mixtures of different damping types are common among oscillators, and only with
viscous or hysteretic damping is the Q independent of amplitude. Other cases
may result, for example, from the decay being a combination of hysteretic
damping and amplitude-dependent damping. An example used to illustrate this
combination was an outgassing pendulum oscillating in
vacuum. Similarly, a long, 'simple'
pendulum, oscillating in air, is found to require a pair of terms--viscous
damping and 'fluid' damping [Nelson 1986].
In the Nelson/Olsson experiment, the drag was found, because of the size
of the Reynolds number, to involve both first and second power velocity terms.
Their case can, incidently, be treated by the
modified Coulomb, generalized damping model of this document.
The
presence of either amplitude-dependent damping or Coulomb damping is expected
to play a role in determining what modes of a many-body system are actually
excited by external forcing. Concerning
the latter, Coulomb friction is the basis for exciting chaotic vibrations in
mechanical systems [Moon 1987]. Without the nonlinear friction, the excitation
would be impossible. In similar manner
(although chaotic motion may be present but not in an obvious way), friction
from rosin on a violin bow is used to play the violin. Still another example of similar physics is
the 'singing rod' that was mentioned elsewhere as exhibiting thermoelastic damping.
Whatever
combination of normal modes are initially excited in a
linear system are the only ones that can exist thereafter. Such is not the
case, however, for many systems; and since nonlinearity is required for mode
coupling, there must be nonlinearity in the equations of motion. There is no question about the existence and
importance of elastic nonlinearity.
Indeed, thermal expansion would be impossible in the absence of higher
order elastic constants. The importance
of nonlinear damping remains yet to be quantified; since models to include it
have been few in number. For those who
have found it advantageous to include the oldest and simplest type of
nonlinearity in a damping model--Coulomb damping (sliding friction); the
improvements realized by their choice are unlikely to cause them to revisit the
problem and try to solve it in terms of a viscous equivalent linear approximation.
There
are many examples of damping of a single type other than viscous. In their
efforts to improve the knowledge of the Newtonian gravitational constant G =
6.67 x 10-11 N m2/kg2 (approx.), Bantel
and Newman discovered a pure form of amplitude dependent damping of internal
friction type [Bantel 2000]. They did their experiments at liquid helium
temperature (4.2 K) and note the following:
"A striking feature noted in our data is the linearity of the
amplitude dependence of Q-1 for the three metal fiber
materials", and also "Linearity implies that Q may depend on
frequency but not on amplitude, while in fact Fig. 1 displays a significant
amplitude dependence (and hence nonlinearity) of internal friction in all
fibers tested." They also considered
the temperature dependence of damping and note that there are two independent
contributions in CuBe. One is linear and temperature
independent and the other amplitude-dependent and independent of
temperature. Finally, it is worth noting
their statement, "...our results are strongly suggestive of some kind of
'stick-slip' mechanism ...", which lends strong
support to the modified Coulomb internal friction damping model of the present
document.
Repetition
is felt to be warranted--such systems cannot always be reasonably described by
an equivalent viscous form! For a case of amplitude dependent Q, the equivalent
form has no meaning unless the amplitude is fixed; i.e
it oscillates at steady state.
Unfortunately, the evolution of the system to steady state is expected
to depend on the damping form(s). Surely a model (not yet realized) that
predicts what modes survive is worth much more than one which only
characterizes the modes after they've reached steady state. The author and
Prof. Dewey Hodges of Georgia Tech's
As
demonstrated by Bantel and Newman, the mixture of
damping types that can co-exist in a system may change with temperature. Early experiments by
One of the remarkable things about the majority of damping models has been the absence of a direct consideration of energy in describing the dissipation process. After all, the most important quantity transformed by the damping is energy; so its inclusion is natural.
18.2
Harmonic Content
When
the damping is nonlinear, the waveform of the oscillator in free-decay contains
harmonics. The harmonic content is most
obvious in the residuals (difference) after fitting a damped sinusoid to the
record, as shown in Fig. 23.
Figure 23. Harmonic differences between the residuals of the modified Coulomb
Damping Model and the classic viscous damping model.
For reference purposes, a pure sawtooth
is included in the figure.
Residuals are still present for the
viscous case because the equation of motion was integrated numerically and
compared against the classic exponentially decaying sinusoid (solution to the
equation) that was used for fitting in all cases. There is always some degree
of mismatch with the fit because of rounding errors in the computer. In Fig. 23 the fundamental is
smaller for the viscous case because the fit is inherently more perfect by
about an order of magnitude in most of the ‘eye-ball’ fits that were performed
by Excel after importation of the data.
A
test for
harmonic content was performed on the seismometer (17 s period) data displayed
in Fig. 11 illustrating phase noise. The
power spectrum of the residuals for that case are
shown in Fig. 24.
Figure 24. Power
spectrum of residuals, Sprengnether vertical
seismometer free decay, showing harmonic content.
The third harmonic is especially noticeable in this case. That the other harmonics are not so 'cleanly' displayed may result from the significant phase noise of the record.
By looking at the FFT of residuals, rather than the experimental record itself, one finds evidence for a combination of both mechanical and electronic noise. At lower frequencies the noise (largely mechanical) is approximately 1/f, while at higher frequencies the noise (largely electronic) begins to be more nearly ‘white’ (frequency independent) because of discretization errors of the resolution-limited 12 bit A to D converter.
In general, more spectral information can be gleaned from a consideration of the residuals than from the experimental data alone, particularly as one looks for harmonic distortion of mechanical type. Spectral ‘fingerprints’ may prove ultimately useful in determining to what extent damping models of engineering type need to be implemented in full nonlinear form as opposed to an ‘equivalent viscous’ form that is more convenient mathematically.
The importance of the harmonics observed in Fig. 24 in determining system evolution is not completely known. It was noted earlier that they are expected to influence the evolution of a many-body system to steady state. Presently, it appears that they may serve to validate damping models. From one model type to another, there can be significant differences in the spectral character of the residuals, as shown in Fig. 25. As compared to Fig. 23, the fit with the modified Coulomb (hysteretic case) model has been 'tweaked' to reduce the fundamental somewhat; but the odd harmonics remain significant. Observe that the spectrum of the residuals is almost the same for this model and the simplified structural model [c.f., de Silva, p. 354]. This is true even though the temporal variation of the friction force is dramatically different for the two, as seen from the lower time traces that were used to obtain the residuals (which are too small to be seen in the graphs).
Figure 25. Illustration of the spectral difference of
the residuals for three different damping models. The corresponding temporal records used to
generate the spectra are also shown underneath each case.
From
this author's perspective, the simplified structual
model is unrealistic, since the friction force, given by
, vanishes for zero displacement (the absolute value of the
displacement being used to get the hysteretic form of frequency
dependence). This is seen in Fig. 26,
which compares hysteresis curves for several
models. The modified Coulomb case shown
is slightly different from equation (52) that was used to generate Fig 25; Fig.
26 was generated with the Aprev shown in
equation (10).
Figure 26. Comparison of hysteresis curves for some damping models.
More studies of this type are obviously called for. The spectrum of residuals is a powerful means for the study of damping physics, and it needs to be more widely employed.
18. 3 Nonlinearity/complexity & Future
Technologies
Nonlinear damping models must improve if we are to overcome various
technological barriers. One barrier is
in the area of civil engineering. One of
the pioneers of finite element modelling (FEM) is
Prof. Emeritus Edward L. Wilson, of the
"Linear viscous damping is a
property of the computer model and is not a property of a real structure".
Expanding upon the statement, he notes: "the use of linear modal damping, as a percentage of critical damping, has been used to approximate the nonlinear behavior of structures. The energy dissipation in real structures is far more complicated and tends to be proportional to displacements rather than proportional to the velocity. The use of approximate "equivalent viscous damping" has little theoretical or experimental justification......the standard "state of the art" assumption of modal damping needs to be re-examined and an alternative approach must be developed." (in reference to Rayleigh damping).
One of the 'hi-tech' areas where modelling improvements are also sorely needed is that involving miniaturized mechanical systems. For example, MEMS devices have already encountered some of the 'strange phenomena' of solid state physics mentioned by Richard Feynman in his famous 1959 talk. To master or compensate for these phenomena, better understanding of the physics will be necessary.
18.4 Microdynamics, Mesomechanics,
& Mesodynamics
At least three different broad fields of research have focused on problems associated with the structural defects that cause hysteresis. These are:
18.5 Microdynamics
In the 'microdynamics' world, the emphasis appears to have been primarily on 'contact' friction. The 6th Microdynamics Workshop held at Jet Propulsion Laboratory in 1999 produced the following statements (quoting Marie Levine's Program Overview): (1) "We have demonstrated that microdynamics exist. The next step is to qualify and quantify microdynamics through rigorous testing and analysis techniques."(2) Microdynamics is "defined as sub-micron nonlinear dynamics of materials, mechanisms (latches, joints, ...) and other interface discontinuities".
In this workshop it was noted that 'frequency-based' computational methods cannot be used to model quasi-static, transient, & non-stationary disturbances. One of the flight operations they have recommended to minimize adverse effects of microdynamics is dithering.
18.6 Mesomechanics
Ostermeyer and Popov have the following to say about mesomechanics: "Real physical objects inherently possess discrete internal structures. Great efforts are needed to formulate continuum models of really granular bodies. The history of the last two centuries in a multitude of ways has been marked by highly successful attempts at formulating and analyzing the continuum models of the discrete world. In spite of great advances of continuum mechanics, a number of physical processes are amenable to simulation within the framework of continuum approaches only to a very limited extent. Among these are primarily all the processes whereby the medium continuity is impaired; ie., those of nucleation and accumulation of damages and cracks and failure of materials and constructions." [Ostermeyer 1999] .
Their paper speaks to one of the difficulties concerning granular materials that was mentioned earlier in this chapter--that the potential energy cannot be defined in the common manner. They introduce a temperature dependent non-equilibrium interaction potential that is not constant in time, due to the relaxation processes occurring in the system.
18.7 Mesodynamics
The author of this chapter is singlehandedly responsible for the use of the term 'mesodynamics'
in the context of mechanical oscillators.
His research has been conducted independent of those doing mesomechanics; he came only recently to know of the
latter. Whereas mesomechanics
seems to have been largely concerned with failure, mesodynamics
has been concerned with low level hysteresis. It is probably closely related to the
aforementioned microdynamics, except that the latter
seems to have focused on surfaces (sliding friction), whereas mesodynamics is concerned with internal friction.
A group of individuals using 'mesodynamics' to describe some of their computational
physics is part of the Materials Science Division of Argonne National
Laboratory. Their description of
computational theory includes: (i) atomic-level
simulation (using molecular dynamics), (ii) mesoscale
simulation; i.e., 'mesodynamics' (using FEM), and
(iii) macroscale (continuum) simulation (FEM). Like the author of this chapter, they
recognize that the mesoscale is not a continuuum (meaning, for example that the foundation of viscoelasticity is for many cases on shaky ground). They employ "dynamical simulation
methods in which the microstructural elements (grain
boundaries and grain junctions) are considered as the fundamental entities
whose dynamical behavior determines microstructural
evolution in space and time."
At the Theoretical Division of Los
Alamos National Laboratory, Brad Lee Holian has been modelling mesodynamics via nonequilibrium molecular-dynamics (NEMD). In his paper, "Mesodynamics
from Atomistics:
A New Route to Hall-Petch", he notes that
(i) the mesoscopic
nonlinear elastic behavior must agree with the atomistic in compression; and
(ii) the mesoscale cold curve in tension represents
surface, rather than bulk cohesion, thereby decreasing inversely with grain
size; ...[Holian, 2003]
The
complexity of mesodynamics, which this author has labeled 'mesoanelastic
complexity' is responsible for much of the aforementioned 'strange
phenomena'. To those familiar with the Barkhausen effect and the PLC effect, they are less
strange. It is thought that Richard
Feynman, if he were still alive, would identify with mesodynamics
because of material in his three-volume series [Feynman 1970]. For example, we've already noted his
discussion of the Barkhausen effect; and he included
in its entirety a reprint of the Bragg/Nye paper on bubbles; which show
two-dimensional defect structures such as dislocations, 'grains', and 'recrystallization' boundaries after stirring. [ Bragg 1947] .
Another famous individual, whose work related in
an unexpected way to the material of this chapter, was Enrico
Fermi. In one of the first dynamics calculations
carried out on a computer, he and colleagues treated a chain of harmonic
oscillators coupled together by a nonlinear term. [ Fermi 1940]. The continuum limit of their model is the
remarkable nonlinear partial differential equation known as the Korteweg-deVries equation; whose solution is a soliton, used to advantage in optical fibers. Damping of solitons, whether of the KdV type
or the Sine Gordon (kink/anti-kink) type, is not to be described by linear
mathematics. Incidently,
the Sine Gordon soliton is used in modelling dislocations [Nabarro
1987]. The earliest theory to describe dislocation damping using kink-antikink pairs was that of Seeger
[Seeger 1956].
18.8 Example of the importance of Mesoanelastic complexity
As noted earlier in this chapter, once hysteretic damping was finally recognized to be important to the Cavendish experiment, better agreement with theory and experiment was possible. Curiously, Henry Cavendish may have been the first person to encounter a 'strange' phenomenon (which he did not discuss) [Cavendish 1798]. In his first mass swing to perturb the balance, which used a 'fiber' made of copper (silvered), there was an anomalously small period of oscillation that was only 55 s. The period reported for subsequent trials was about 421 s.
Whereas the Michell-Cavendish apparatus was a torsion balance, the instrument of Fig. 26 is a physical pendulum. The perturbing masses (M), were hung from a bicycle wheel whose axle was suspended from the ceiling. The long-period pendulum was placed under a bell jar so that the instrument would not be driven by air currents. By rotating the wheel at constant angular velocity, the driving force on the pendulum was harmonic. (In the figure the position of the M's one-half period later are shown by the dashed cicrcles.) Knowing the amount of damping, as determined from large amplitude free-decay, it was easy to estimate the number of orbits of the bell jar, at the resonance frequency of the pendulum, required to excite motion to a level above noise in the sensor. Surprisingly, if it was initially at rest, no amount of drive by this means was able to get the pendulum oscillating! The reason involves metastabilities of the defect structures. The potential well is not harmonic (parabolic), but is rather modulated by 'fine structure'. When located in a deep metastability, the small gravitational force of the drive (in nanoNewtons) is not able to 'unlatch' the system. If the pendulum had been dithered (a practice used in engineering) this problem could have been at least partly avoided. As it was, the pendulum rested on an isolation table of the type used in optics experiments.
Figure 27.
Physical pendulum used in the late 1980's to try and measure the
Newtonian gravitational constant.
More recently, a Hungarian research team has used a similar apparatus
and postulated that the anomalies of their experiment derive from gravity being
other than prescribed by
19. Concluding remark
Much of the material of this part of the damping chapter is clearly not applicable to direct engineering application. It was deemed important to present some of the extensive background information responsible for birthing the practical equations of section 17. In PART II which follows, the reader will find practical aids to the measurement of damping.